My understanding is that it's a Latin idiom for stating someone's name, like in French: "Je m'appelle Jason". No part of that literally means "the name belonging to me" so as to necessitate a possessive adjective. Likewise Icelandic "Ég heiti Jason"; both of these essentially mean "I am called Jason".
However, I'm willing to be corrected if you are saying that this is in fact the incorrect idiom for Latin, because you are a classicist, or someone who knows better.
That would be a correct way to express possession! However, there are several ways to express possession in Latin, and using the dative (mihi) is one of them. It's called the "dative of possession". In English a sentence like "The name is to me" (in Latin, "Mihi nomen est") doesn't make much sense, but it's a sort of idiomatic construction in Latin. We would usually translate that as "It is my name" or "The name is mine" to make it easier to understand. Hope this helps!