Sora did not just use "Surname" to fill in the blanks of her fake Imperium ID. It's a placeholder, until she found a true family who calls her their own. A family she gets to choose for herself, after being stuck with parents who did not love her as their own.
Perhaps that's why her actual last name --- her parents' last name --- has not been revealed, even after four seasons of the show. Just as she was seen as worthless by them, her last name is worthless, therefore the knowledge of it would be worthless to us, the audience.
But that doesn't answer why she uses "Surname." Why would she need a placeholder as a last name if she could just come up with a real one for herself? She has the freedom and means to do so, anyway.
At the time she filled in the information for her fake ID, she was struggling with her worth, who she was, what she's capable of. Throughout season one, she tried to solve her inability to harness her Elemental powers by herself as if that could prove her own worth, yet when it came down to it, when she finally confronted her parents after all those years, she realized that that didn't matter. She already had an abundant amount of worth. What mattered were the people in her life who reminded her of that worth. Those are the people she will choose as her family.
She has made her choices, yes, excellent choices, but the only person who she has consistently, persistently chosen, was Arin Nived. He reminded her by supporting her, by reassuring her, by staying with her, by choosing her over and over again. He could've left her on the caged trunk of the truck that was supposed to carry her away, and said goodbye to their friendship. But he didn't. He reminded her yet again by rescuing her, then sealing their friendship in an everlasting vow.
She doesn't need to give herself a new name --- she already did it for Ana. She has always been a Nived (although using the name itself would imply...certain things, but I digress).
Perhaps she needed to realize that, too. Perhaps she needed time to come to terms with it, to finally accept not just his family name, but the love, warmth, and companionship that came with it, hence the use of "Surname" as a placeholder. And perhaps, by "Surname," she was giving him time to realize and accept it as well.
In due time, he does. He calls her as a part of his family, just as she did when she earned her True Potential, and saved his life.
