That is all beside the point. Yes, you are given a name, else you would be nameless until you understood the concept of names. That doesn't mean you have to keep that name. You can change your last name too. Women do it all the time when they get married. As of recent years, some men change their name to match their fiancé's last name. People change their name across languages when they immigrate. My girlfriend is adopted and she is planning on using her adoptive family's last name for the rest of her life even though she is becoming reacquainted with her biological family. Her biological family named her, not her adoptive family, but it's her choice. Last name, first name, middle name, second middle name, whatever - it's all a choice.
If Leelah wanted to ditch her family, which would not surprise me given how they treated her, she could have gone with a different name entirely. As before, her legal name on her medical and insurance forms or whatever would have remained the same, but she was under no obligation to retain her last name either, nor is anyone else.
"the name you get called each day is chosen by your friends"
I literally addressed exactly that in my posts before you. Did you bother reading? My sister tells her friends what to call her, because she doesn't go by her name. When her professors read off her name for attendance, she corrects them. She chooses her name. When my dad, who is struggling to adjust to her new name, calls her by her birth name, she corrects him too. Octavious, if your friends are choosing to call you something you don't want to be called and you tell them that yet they still continue, you would have some really shitty friends and ought to get away from them.