Early Italian Naming Convention

In the 1400’s, at least in Italy, it was apparently commonplace to name someone based on where they were born, hence Leonardo da Vinci (Leonardo from the town of Vinci), but how was this naming convention practical? What if there were two Leonardos who happen to have been born in Vinci at the same time? Would only one of them have had the ‘da Vinci’ last name and the other a regular last name? And if Leonardo had married would his son have been, for example, Antonio da Vinci even if he was born in Florence? How did this all work in real life?

Can I get a cookie for knowing who it would be about?

I’ve been searching for a recent thread on the subject which was well-documented (someone was actually looking up information on tax records of the time), but the search function is being nasty and giving me a blank screen. The conclusion of the thread was that “da Vinci” was his actual family name, not merely a personal one. So yes, his son Antonio would have been a da Vinci without being born in Vinci, but this is the same as the fact that anybody called Smith nowadays is highly unlikely to know how to work a forge.

Now to answer “what would happen if two people were born in the same place and had the same firstname”:
The nicknames which eventually become lastnames spring from the need to differentiate people; if the “da Vinci” had not been his family name and depending on circumstances, two Leos born there close together could both have ended up being called “da Vinci” (if both moved away to different places, in which case the nicks would stem from the need to differentiate them from other people and not from each other), or they could have ended up with completely different nicks: say that both stayed in town and got different jobs or, like these classmates of mine who had the exact same first and lastnames, they would have gotten nicks based on physical characteristics (in my classmates’ case, “blonde” and “dark haired”).