Let's put this another way. Why aren't all Clerics Acolytes? Why aren't all Wizards Sages?
The Class identity and the things you supposedly would have to have done to gain your Class powers generally points towards this being the most likely origin...but you can be a Street Urchin Cleric and a Noble Wizard just as easily.
"Well", one can say, your Background was what you did before you got your Class, and it's in addition to your Class identity.
But when a Background and a class coincide in identity, you run into some issues. Is that overly-litigious Abbot a Cleric? A Monk? A Rogue with the Acolyte Background? If he's an NPC, he doesn't have either a Background or a Class, and his mechanics simply reflect what he needs to be, which might not even include skills!
Thus it's just as easy (and possibly as correct) to say a medieval warrior-priest could be a:
*Cleric, quite easily one with a Noble background if he's the second son of a prominent family and thus could easily ascend to the highest ranks of the Church.
*Fighter with the Acolyte background.
*Paladin with either.
*Or something else entirely.
Where in previous editions, you could naturally assume an Archbishop is a high level Cleric, there's no longer any reason to assume that's the case. Maybe he's an Acolyte Warlock using his Pact powers to pretend to be a deeply holy man! Who can say?
This means that the "Class Identity" is somewhat irrelevant, even if the "Class Role" mostly exists- though in 5e, the game takes pains to make a particular Class Role as vestigial as possible, since players value freedom, and don't want to feel forced like they are being sorted into a preset role. Thus you have Fighters who only care about damage and don't want to protect people, Wizards armored like walking tanks, and Clerics who only cast Spirit Guardians!