I think it's very unlikely that Quirrel in HPMoR is THE Quirrel, since the basic biographical details given of Quirrel's time at Hogwarts line up with those given for the canon Quirrel. I think we can take both the Aurors and Professor Quirrel's assertions at face value on that score.

It seems significant to me that in HPMoR no one has mentioned Quirrel's previous tenure as Professor of Muggle Studies-- they all appear to act as though they didn't know him before his term as Defense Professor. This suggests to me that the original Quirrel has in fact been missing for some time and never actually became a Muggle Studies professor at Hogwarts. This being the case, it would be relatively simple for someone to take on Quirrel's name, and he likely wouldn't even have to LOOK like the original Quirrel.

As to who Quirrel is, I think perhaps Moody has the closest hypothesis: the John Monroe identity is probably adopted as well, and the Monroe that people remembered from the Wizarding War was never Monroe himself. This forms a pattern of a powerful wizard identifying missing and possibly dead wizards (or causing that state himself), taking their identities, and using those identities to act in current events. We know from The Incident with Rescuing Bellatrix from Azkaban that the person currently calling himself Quirrel has many other identities, so it's likely that John Monroe was also not the first.

Moody's Three Types of Dark Wizards would imply, then, that Voldemort is the second type while Quirrel is the third type.

As for who this person originally was, I have to wonder if John Monroe was the first identity he adopted. If he actually IS the original John Monroe, then he was born in the 40s and is therefore in his 50s at the time of the present story. If he isn't, then he was born before that and it begins to stretch the imagination that he might be able to pass himself off as being in his late 30s, even considering the longevity of wizards. Granted, while he doesn't have to be a Metamorphmagus to adopt new identities, it would certainly help while also solving the age problem.

Another possibility might be that our serial identity thief might actually have the same identity as another wizard we've heard of who likes to adopt new identities, as hinted in the book Hermione was reading when she was trying to find a way to get Harry's debt absolved. Were Quirrel to be Nicolas Flamel, it might answer quite a lot of questions, though there are some problems with the idea. Dumbledore is well-acquainted with Flamel and appears to be maintaining contact with him (which would be easy to do, granted, if Quirrel is Flamel) but doesn't show any signs of knowing that Quirrel is Flamel.