I tried to login but the screen showed me to make a family name. I already have a family name "루나" with the account. It doesn't let me login.

Just change the server. On You might be playing on Europe (Asia/America) server, but there is more of these.
For example: I was playing on Velia(Europe), but didn't notice that I chose Balenos(Europe).
Just change it and play on 😉.