What is a display name on a profile?
A Display Name is how you want to be known to the community. It can be different from your real name. It is often consisted of a first name and potentially last name. A Username is a name unique to you, and it is how the community can reference you specifically in their conversations, similar to Twitter or Instagram.
Can I ask for your full name?
The simple way to ask someone's name is to say, "What is your name?" If you are concerned that a question -- any question -- is abrupt or potentially rude, you can soften it by saying, "Excuse me, but ..." or "Would you please tell me ..." or "Would you mind telling me ...".