The phrase एकं सद्विप्राः बहुधा वदन्ति (Ekam sad vipra bahudhā vadanti) is popularly taken to indicate some sort of Vedic egalitarianism of all deities as being manifestations of One Being. Sayana's commentary gives a similar secular interpretation that Aditya is called Indra, Mitra etc. because they are identical in essence. However, a scrutiny into the verse reveals that such an interpretation is textually and logically wrong.
The full verse (Rgveda 1.164.46) runs thus:
इन्द्रं मित्रं वरुणं अग्निं आहुः अथो स दिव्यो सुपर्णो गरुत्मान् |
एकं सद्विप्राः बहुधा वदन्ति अग्निं यमं मातरिश्वानमाहुः ||
Shrimad Anandatirtha says the above verse conveys that One Being i.e. Vishnu possesses the names of all devatas. While commenting on the Aitareya Upanishad's सर्वा ऋचः सर्वे वेदाः... प्राण एव (which tells how different Rishi names convey the same Paramatma), Acharya says - thus, names of all rishis are primarily referrents of Vishnu. What then to say of names of Devatas! Acharya then quotes the line यो देवानां नामधा एक एव', the verse under consideration (indram mitraM) and a couple of others are quoted to show how One Being is the referrent of all names. There is a more detailed presentation in other parts of Aitareya bhashya (multiple places including brihati sahasra context) and Anuvyakhyana 1.4.
Broadly, there are 3 ideas on why and how names of all devatas refer to Parabrahma Vishnu.
The Abheda / Identity position: All devatas are identical in essence with One Being.
The individual devata may have some non-essential pecularities, owing to some artificial boundary but is essentially the same as the One Being. For example, the space inside a pot is bound, but is essentially identical with space elsewhere. This position is that of advaitins in general, and Sayana in his commentary on the above Rk.
The Antaryami / Inner Being position: One Being is present inside all devatas.
The references of the devatas' names does not _stop_ with the devata but 'extend' to the Being present inside that Deity, to the Being whose mode is devata. This is just like normal references is to the indweller, not just the body. The antaryAmi position is that of Vishishtadvaitins in general. It is explained by Ramanuja in his Vedartha sangraha (among other works): अश्रुत-वेदान्ताः पुरुषाः पदार्थाः सर्वे जीवात्मनश्च ब्रह्मात्मका इति न पश्यति सर्व-शब्दानां च केवलेषु तत्तत्पदार्थेषु वाच्यैकदेशेषु वाच्यपर्यवसानं मन्यन्ते । इदानीं वेदान्त-वाक्य-श्रवणेन ब्रह्मकार्यतया तदन्तर्यामितया च सर्वस्य ब्रह्मात्मकत्वं सर्व-शब्दानां तत्तत्प्रकार-संस्थित-ब्रह्मवाचित्वं च जानन्ति
The Vaachyatva / Direct referrent position: One Being possessing the names of all devatas.
The Vachyatva position is that the names of all devatas belong to One Being. A name refers to an entity if an entity possesses that quality which is indicated by that name (or word). For example, 'Indra' refers to the quality of having 'parama-aishvarya' (High overlordship / prosperity/ capability to make things happen, undo them or make them happen differently). A person is called Indra if that person possesses that quality. OR that quality, though is not possessed by that person, is under that person's control (like a person is called wealthy because though the money may not with that person then, it is still under that person's control). This is told in the famous statement in Anuvyakhyana: स्वातन्त्र्यं तद्गतत्वं च शब्दवृत्तेर्हि कारणम् । स्वातन्त्र्यं तत्र मुख्यं स्यात् . It must be highlighted that dvaita does accept the antaryAmi idea, as can be seen in Acharya's bhAShya on Brahmasutra 1.4.1: that word avyakta refers to Prakriti and its inner Being. But we reject the 'brahmAtmakatva' idea (that everything is a material-variant of ParaBrahman just like pot is of mud).
Shri Vyasaraja rejects the abheda position (Sayana's position) in his Tatparya-chandrika (in the siddhaanta portion of अन्तस्थत्वाधिकरण 1.1) and presents Acharya's view in a consolidated manner. He says -
The word आहुः clearly indicates being called / referred / addressed by those names. That is, One Being is called as Indra, Mitra, Varuna etc. This is the direct meaning. The verse does not say Indra etc. and other devatas are all essentially One.
The same verse subsequently posits difference between Vishnu (that ekam sat) and others -- सप्तार्धगर्भा भुवनस्य रेतो विष्णोस्तिष्ठन्ति प्रदिशा विधर्मणि | This is explained by Shri Raghavendra tirtha thus - भुवनस्य रेतः - रतिप्रदः विष्णोः प्रदिशा आज्ञया अर्धगर्भाः आसमन्तात् ऋद्धं ब्रह्माण्डं तद्गर्भाणि सप्त-महाभूतमहदहंकार-तत्त्वानि तदभिमानिनः | विधर्मणि विविधधारणकर्मणि | तिष्ठान्ति इति विष्णुना भेदोक्तेः | Upon on the command of Vishnu, who gives joy to the universe, the devatas controlling the seven tattva (5 panchabhutas + mahat + ahankara tattva) of this well prosperous brahmanda, indulge in their various activities. The idea is these devatas are said to be विष्णोः प्रदिशा i.e. under his command, thereby indicating a difference between that One being and different devatas.
The Shruti 'यो देवानां नामधा एक एव' (Rigveda 10.82.3) - means One possessor of names of the Gods, is in agreement with this interpretation.
He then quotes shruti passages that seemingly talk of identity between different devatas, shows a problem with the identity interpretation and how vAchyatva interpretation avoids that problem.
In the Shruti - सोंंऽतकः स रुद्रः स प्राणभृत्स प्राणनायकः स ईशो यो हरिः (quoted in Acharya's bhashya on Brahmasutra 1.1.3). If identity between Rudra and Hari is intended, the usage of 'sa rudraH' and 'sa IshaH' makes for an unnecessary repetition (punarukti). It makes sense only if vAchyatva i.e. being referred by those names is meant.
The shruti 'स संकृतिः विश्वकर्मा स प्रथमो मित्रो वरुणो अग्निः | स प्रथमो बृहस्पतिः' (taittariya brahmana 1.1.1.5) - which means He is the 'first' vishvakarma, first mitra, first varuNa, first agni, first brihaspati - indicates that there is yet another vishvakarma, mitra etc., without which the adjective 'प्रथमः' becomes redundant. Thus, even here, there is a difference between the devatas and the primary possessor of those devata names.
In the Shruti 'यं इन्द्रं आहुः वरुणं यं आहुः यं मित्रं आहुः यं उ सत्यं आहुः' (taittariya brahmana 3.7.9.98), the apparent identity interpretation is contradicted by the next line - यो देवानां देवतमः तपोजाः तस्मै त्वा तेभ्यः त्वा'. This talks of that Being who is the supreme most among devatas (devatama) and also about offering oblations to that Being (tasmai) and those devatas (tebhyaH) separately. More details on this can be seen in Acharya's bhashya on atharvaNa upanishat (mundaka).
Ofcourse, shrutis such as 'भीषास्माद्वातः पवते', 'भयादग्निस्तपते' which indicate that devatas such as agni, vayu, indra, yama function out of fear (of causing displeasure to their lovable Lord) - very clearly posit difference between devatas and that Being who possesses their names.
In fact, there is no dearth of shrutis that indicate difference between the devatas. After all, devatas play different roles in yajna, or some other event. So passages conveying these roles imply difference between devatas. However, somebody like Sayana may attempt to undermine these statements as transactional / empirical (व्यावहारिक) and not of final nature. And they use statements such as 'एकं सत् विप्रा बहुधा वदन्ति' to justify undermining the difference-statements. Hence Acharya and his meticulous followers like Shri Vyasaraja show a problem (बाधक) in the identity interpretation of those statements.
There is an additional problem in 'एकं सत् विप्रा बहुधा वदन्ति'. The word 'agni' is mentioned twice. The identity-interpretation makes that redundant. Sayana brings in an unmentioned adjective to interpret the second agni as 'वैद्युत अग्नि'. While this is not a serious problem by itself, it could have been avoided with the vAchyatva interpretation. Yaska interprets the first Agni as the uddeshya i.e. that ekam sat agni - is called as Indra, agni etc. thereby not getting over redundancy.
In the vAchyatva interpretation, the two mentions of agni indicate two different प्रवृत्ति-निमित्त i.e. two different qualities indicated by the word agni. Thus the Supreme Being (that One Being) is called / referred by names Indra (because of His extreme opulence), mitraM (because He measures and protects), agni (because He leads from the front), and agni (because He carries the body and this world) etc. Shri Vyasaraja also points out that 'vipra vadanti' clearly supports the tenet of विद्वद्-रूढि Vidvadrūḍhi (see p.s.)
One can notice that the same guideline can be applied to such 'apparent identity' statements across the scriptural spectrum. For example, the following statements must be interpreted in terms of vAchyatva i.e. the Supreme Being being the referent of those names (because He has those qualities indicated by those names):
Mahanarayana Upanishat : स ब्रह्मा स शिवः स हरिः सेन्द्रः सो अक्षरः परमः स्वराट् indicates the paramaatma being vAchya i.e. referred by the names Brahma, Shiva, Hari, Indra, Akshara. Elsewhere in Chandrika, Shri Vyasaraja points out the identity interpretation leads to the problem of Paramata being identical to an insentient letter (Akshara). For, the word 'Akshara' refers to the letter in rUDhyartha. If Akshara has to be interpreted in its etymological sense as 'indestructible', the same applies to other names mentioned there. (So much for the debate between Advaitins and Vishishtaadvaitins on the presence or absence of sa hariH in their paaTha).
Mahanarayana Upanishat: यज्ञस्त्वं वषट्कारस्त्वं रुद्रस्त्वं विष्णुस्त्वं ब्रह्म त्वं प्रजापतिः - here too the identity-interpretation implies Paramatma is identical to insentient stuff like yajna etc. Only vAchyatva interpretation makes sense.
Thus, the identity-interpretation doesn't pay attention to details and is flawed. On the other hand, Shrimad Acharya's vaachyatva-interpretation is logically tenable and is in line with other statements such as 'यो देवेषु अधिदेवः एकः' or the Harivamsha and Mahabharata verses that state all Vedas to be conveying Shri Hari's names primarily.
The other theory holds that all names refer to Parabrahman because He is the indweller of all beings. Words like manuShya, deva refer not just to the body but to the indweller / the qualified whole. Similarly, all words refer to the indweller / the qualified whole. This idea is refuted by Shri Vyasatirtha towards the end of chandrika 1st adhyaya. He points out words like sharira, jaDa, thin, fat stop at the body, and do not extend to the indweller / qualified whole. Words like soul etc. refer to the indweller only and not to the body / qualifier. Similarly in the shrutis such as 'AkAshAdeva samutpadyante' (all this is born out of Akaasha), the word 'AkAsha' refers only to Parabrahman, not to the material AkAsha at all. On the other hand, the same word in 'AtmanaH AkAshaH sambhUtaH' refers only to the material AkAsha and not to the indweller at all. So it is incorrect to assume that all words extend to the indweller of things or the qualified whole. He also gives instances to demonstrate how this idea that words refer to the qualified whole leads to illogical situations. For example, the shruti 'tat sriShTvA tadevanuprAvishat' has the Ishvara entering an entity that already has the Ishvara in it.
Most fundamentally, as pointed out elsewhere, this approach does not lead to asserting a positive attribute in Brahman. For example, Brahman is called Akasha because He is the indweller only. On the other hand, the vAchyatva interpretation asserts that Brahman is called Akasha because he possesses the quality of being radiant (AkAsha is derived from kaash diptau) and in unsurpassable measure, and is the main controller of that quality wherever it is seen (irrespective of the location of the controller).
श्रीमध्वेशकृष्णार्पणमस्तु
p.s.
Vidvadrūḍhi is the conventional notion among vidvaan (i.e. the knowledgeable). Other schools of thought hold that rUDhi - the conventional meaning which is generally used by public at large - suggests the primary referrent and it should be given up only in cases of conflict or contradiction with a stronger source. Dvaita accepts that but holds that there are 2 such rUDhis. One is the conventional word-meaning mapping which is prevalent among the public. The other is the convention of word-object meaning among the knowledgeable - vidvaans. This rUDhi - convention among experts (if one may) is stronger than the conventional meaning held by the not-knowledgable public. We can observe this rule in daily life too - the word stress means differently to public and to physicists. When studying Physics, the technical meaning (prevalent among physicists) is preferred over that meaning prevalent in public. This idea of Vidvadrūḍhi is upheld by the Veda itself, where it uses phrases like 'viprA vadanti' (the knowledgeable refer to), तदु सत्यं आहुः तदेव ब्रह्म परमं कवीनाम् (the wise people refer to the Being on the Ocean as Parabrahman etc.), शतर्चिन इत्याचक्षत एतमेव (This Being alone, not the Rishi with that name, is referred primarily as shatarchin).