All the characters who live in Zion have names using a similar convention, and this appears to be the only names they use there. On the other hand, inside the Matrix, names work just as they do in our reality (or maybe our reality is the Matrix...).
In the first movie, we have two very clear examples:
Firstly, as you point out, the protagonist grew up inside the simulation of 20th Century America, named according to that society's convention "Thomas A. Anderson". Then when he chose the red pill, he was introduced to the crew of the Nebuchadnezzar simply as "Neo" - a name he had already chosen in other contexts, in his case illegal hacking.
Secondly, we meet characters who were born in Zion, who have no previous identity in the Matrix:
Neo: You don't... you don't have any...
Tank: Holes? Nope. Me and my brother Dozer, we're both one hundred percent pure, old fashioned, home-grown human, born free right here in the real world. Genuine child of Zion.
There is no hint that Tank also has a "traditional European name", with a first name and a surname; that's just not how names work in Zion.
It's reasonable to assume that other people living in Zion had a similar experience to Neo: they had names appropriate to the in-Matrix America, or India, or Spain; then they were freed, and chose new names more appropriate for use in Zion, probably based on aliases they'd used previously (just like a lot of users on this site have aliases; there's no need for illegal activity to be involved).