Curate, connect, and discover
I have a theory about why Chuuya goes by his first name, as opposed to most other people. Because even in the manga, where characters like Fyodor, Nikolai, and Lucy are generally referred to by their surnames, Chuuya goes by his given name in both the manga and anime. *Stormbringer Spoilers*
First of all, when he was fifteen, he likely went by his first name because he was surrounded by friends, by the people who raised him, and going by something like a family name would be distant and unnatural, and likely not something Chuuya would push. So when he met Dazai, he was still immature enough to go only by his given name, and he probably doesn't bother enforcing it when he joined the Mafia either.
But. After the events of Stormbringer and when Chuuya grows up, I suspect it was for a different reason. We don't see much of Chuuya after he escapes the military facility, but since he knew how to speak, my guess would be that he knew enough to know his name, even if he couldn't remember much else. Chuuya was his name, given to him by birth, but the name "Nakahara" isn't. That's N's name, the name of the person who kidnapped him, experimented on him, and tortured him. The real Chuuya's surname was actually Kensuke, and I think something similar applies to him in universe as well. When he was taken by N, the name Nakahara was given to him, but the name Chuuya is the one his real parents gave to him, the name he received when he was born, the name that proves his humanity. So he probably chooses to stick with that name, even though he really could insist on being called "Nakahara" because "Chuuya" is the one that's his.