One of the strengths of British and "Commonwealth" armies is the Regimental system, which serves as a conduit for unit traditions, often hundreds of years' worth, passing down to modern-day and contemporary units - even wartime-only units created during national emergencies.
This system started in the British Army in the 1800s, and had been passed down to armies whose traditions descended from the British, including:
Australia
Canada
India
Pakistan
New Zealand
South Africa
And, of course, the United Kingdom.
In the British tradition, the infantry "Regiment" is not a combat unit. It is, rather, a:
Repository of traditions - flags, battle honors, and other bits of lore, passed down to generations of new members.
Built-in career path for enlisted men, officers and NCOS - the specifics may vary by nation and branch of service
The actual combat units are "battalions" raised under the regiment's auspices. Each regiment may be the parent of several battalions of troops - active-duty, reserve, and units whose traditions exist for mobilization only.
Thus, a British soldier might be a member of the "Second Battalion of the Royal Scots", or the "First Battalion of the Rifle Regiment"; a Canadian soldier, the "2nd Battalion of the Princess Patricia's Light Infantry", and so on.
Cavalry regiments in the British and Commonwealth armies have all the traditional and symbolic aspects of the Infantry regiments (above), plus are also combat units.
Thus, the "Blues and Royals" are both a battalion-sized combat unit and a storehouse of traditions and lore.
Note - British Guards regiments since roughly the 1970s also exist on a 1:1 basis with their combat battalions - although during World War II the Foot Guards (and Royal Armoured Corps and Royal Tank Regiment) units raised separate battalions under the same regimental title. This would not apply in either the Cold War or Modern universes, and is included purely for historical purposes.
Additional note: the Special Air Service is descended from a cavalry unit, and thus its three battalion-sized units are each considered regiments.