I can very easily tell Bronx from Brooklyn from Manhattan from Queens, in some cases down to specific neighborhoods or cultures...but really only in folks 60+, maybe occasionally in people more in the 40+ range who had fairly isolated upbringings, but that's it. Any younger than that, I often can only even tell you're from New York City by word choice and slang, not accent. I suspect that's similar to what's going on in Paris, albeit I'm in absolutely no position to know.

(For context: I did not grow up in NYC, but my family was from there, and I lived there for quite awhile myself as well. It's entirely possible someone who did grow up in NYC can do better than I can, but, at least to my ear, it gets very close to just standard American broadcast English the younger you go, so I doubt it.)

The usual caveat about Paris applies: the city of Paris is but a tiny core of the larger Paris metropolitan area (Paris: 2.1M people, Paris metro: c. 10M). Paris is about one fifth of its metro area, a city like NYC is about half.

A closer analogue to being able to distinguish accents from Montparnasse, Montmartre and La Villette would be to able to distinguish accents from different places in Manhattan.

I think even today there are substantial accent differences within the Paris metropolitan area (think Versailles vs le Marais vs Aulnay), but within Paris proper not so much.