Some of the alters in my system were very secretive about names and didn't like anyone to know them (including other alters). Some of those would also be very secretive about themselves, and some would pretend to be other alters (kind of hiding behind them). Others were very outspoken about their names and separate self-identity. A couple of young alters didn't have a name at all and were glad to be gifted with one. Some fragment parts didn't have a name and didn't require one. So there were lots of different scenarios in our system.
Whatever each individual alters feeling was about having / not-having or owning / hiding a name, it always eventually made sense as to why that was when understood in the context of their individual experiences and history.