How do you determine which FirstName LastName is your ancestor?

NYC census records show my great-grandfather was born in Canada (maybe Montréal?), probably in 1871 or 1872. He might have come to the U.S. as a child or young adult, and there are records of like 36 people born with his name around his birth year in Quebec -- and he might have been born elsewhere. The dates on the records on Ancestry.com vary wildly.

How do you figure out which person with that name is your ancestor? How do you prove it in a way that may persuade on a Canadian citizenship application?