Yes? Yes. America is both. The American political system can be described as a democracy. And a republic. We can call it a federal republican constitutional liberal representative democracy. So why is the phrase "America is a republic, not a democracy" popping up everywhere?
Internationally, if you say republic, it means you don't have a monarch. No more, no less. It does not tell us how you replaced that monarch.
The founders used that word more or less interchangeably, but Madison defined republic as representative democracy where we vote on representatives to vote on things in contrast to direct democracy where we vote on the things themselves directly.
Let's start at why the confusion exists, what words mean, and why (possibly) you are seeing this more now.
James Madison, The Federalist Papers and the start of the Republic vs. Democracy.
Democracy comes from classical Greek, Republic comes from Latin. In the 18th century they were used interchangeably; however, Madison, in the Federalist papers decided to define republic as representative government to distinguish it from direct democracy because the system in the Constitution seemed more akin to Rome with it's anti-king bias and (somewhat) elected senate to represent the people, or to Florence and Venice where people voted on the leaders to represent them.
Contrast this with the ancient Greek system, where people voted directly on laws, officials, and policy rather than representatives to do it for them. (Yes, both of these were highly restrictive, very few people had the franchise, elections were not free nor fair in Rome etc. etc.) In the federalist papers, Madison is making a distinction for readers that grew up in a time and place where this context is widely known to explain why direct democracy is a pragmatic and ideological problem in the US and why representative (republican) democracy can solve it.
One of the pragmatic issue is size. If you are reading this in New England, imagine a town hall for everything. Or if you are in California, proposition votes for everything. Now extend that out to the entire country.
The ideological issue is that while the founders were worried about tyranny from the top, they were also very worried about the "mob" and giving power to the people. The founders were a minority; they were educated, wealthy, landowners, and were worried about all the non-wealthy land owning people getting together and realizing they could outvote the smaller upper class. They were also worried about education, afraid of the non-educated being easily swayed. Since they were outnumbered, this idea that their rights could be voted away by a majority was big concern.
These concerns may have changed slightly in details, but they are still very relevant. How do we prevent a large majority from voting on minority rights? How do we practically carry out government of the people in practice given we are countries of millions?
So these concerns still exist, but the way we define the terms republican, representative, democracy, constitutional, all have changed.
2) Defining the terms today.
As definitions became more solid, you see the term republic refer to countries without hereditary rulers. Spain when it had no king in the early 20th century? Republic. Spain now, with a monarchy? Not republic. China? Well, there's no hereditary rulers, but it's most certainly not a democracy. Think of the IRA in Northern Ireland; that R does not stand for the grand old party.
Democracy is seen as being made up of two main options: direct vs representative. A representative democracy means you vote for people to vote/represent you. You vote for a senator, representative, governor etc? That's a representative, and not direct democracy (it's also why getting rid of the Electoral College won't stop us from being either a representative government (or republican even under Madison's definition: you are voting for a representative either way!)
Direct democracy is when you vote directly on issues, skipping the voting on someone else to vote for you. The US, while mainly utilizing representative government, uses direct government as well: town meetings, recalls, ballot questions, propositions -- basically anytime you vote on a policy or a law, that is a direct democracy.
liberal/constitutional Basically, remember that concern about the majority voting out rights? This is how you prevent that. Constitutions set up the rules of the game and can be democratic or non-democratic. Liberal systems (little l) are systems that protect individual rights from both the majority of the population and by the state/government itself. Liberal here does not mean left-wing, but comes from liberalism and the enlightenment.
Federal means that the constitution establishes certain political sub-units, and gives them powers that cannot be removed by the national level. In the US, the federal/national government cannot abolish a state by an act of legislation; that state is established by the Constitution, just like Congress, and given certain rights and prerogatives. In unitary systems, only the national system is established by the constitution. The national level government can devolve powers down (as in the UK) but can take them away easily. (Bonus: a confederacy is an "alliance of friends" more or less. The EU is very similar to this idea. Here the power is flipped from the unitary system.)
Please note: in the US sub-regional units are referred to as states, in political science and elsewhere the term state typically means the country, regime is the type of governing system, and government means the current political power (in the US, we use administration so internationally, you'd see Trump government instead of the Trump administration). Nation refers to a group united by ethnicity, history, language etc. (remember nationalism?). Please note, that many political scientists that focus on American politics only use the popular US definitions; I am a comparativist and American, so I mix up my american colloquial definitions and polisci all the time, to better confuse everyone-- see my use of representative government up above.
3) Why is this an issue?
Partisan politics. I mean, it could be something else as I lack the power to know men's hearts. But considering this argument is mainly coming from people associated with the Republican party, this seems to be supported.
4) Where to learn more:
WBUR On Point: 2 professors that elaborate on the above, plus an advocate for the republic, not a democracy side. Audio here
NPR. Is America a democracy or a republic? Yes, it is Written overview of the history and Mike Lee's defense of the term republic.
Civics 101: A Podcast Are we a Democracy? A Republic . 2 professors discuss the issue; you can find the transcript and the podcast at the link.
Academic Works:
Dahl, Robert A.., Shapiro, Ian. On Democracy. United Kingdom: Yale University Press, 2020. pages 16-17
Seth Cotlar, “Languages of Democracy in America from the Revolution to the Election of 1800,” in Mark Philp and Joanna Innes, eds., Re-Imagining Democracy in the Age of Revolutions: America, France, Britain, Ireland 1750-1850 (Oxford: Oxford University Press, 2013), 18.