It's not purely a matter of having a name or not, a key part of it is someone having agency over it. Anna chooses to keep going to Nina. People who live under assumed names to avoid persecution or death make the choice to do so. (It's not a very nice choice to have to make, but it is still a choice.) Anonymity and pseudonyms are another angle on this. If someone chooses not to use their real name or any name at all in a situation, that's fine and up to them. In certain instances, no identity or just a number isn't really a big deal; taking a number and waiting in line at a shop, for example; it doesn't really matter, and it's just short term.
But if someone takes you aside, tells you that your name isn't your name anymore and you have to either have no identity or one they force on you, on a permanent or long-term basis and enforce it, it's gonna be fucking unpleasant for you. Maybe not immediately, but it will be.
