I'll answer your second question first. No, this name isn't really used elsewhere--just with your Apple Developer account. If you connect with other team members, they will also see this name.
Now for the first and more important question: Per Apple Developer Support (I called them because I also have this problem), whatever the name on your account was originally when the account was created, that is what your Apple developer name will be forevermore (even if you change your name on your Apple ID). My account was created nearly 15 years ago by one of my parents, so my mother's name is my Apple Developer name (which, obviously, I hate--nothing against my mom). Apple Support was not able to change this name for me.
Here is what you can do:
- Create a new Apple ID with a different email. As tempting as it might be to use the name of your mom, your dog, your neighbor's cat, or Qui-Gon Jinn, use your own name.
- Sign up for an Apple Developer account
- Sign into Xcode with the new account
It's not the cleanest or most convenient solution, but it is a solution (and the only solution, at that).