I read about trademarks that a same name can be reused if the field of usage is different enough.

For example, If I would name a game "The Rising Sun" and it would happen that a music band is also named "The Rising Sun" with a trademark on it, is there any problem with that? Since one is a video game and the other is music, is that a big enough distinction to not cause any legal problem?

That is an excellent question for an IP lawyer.

As far as my training goes, that question is ultimately up to a court. People can speculate, and lawyers can argue either way, but ultimately judges and juries make that decision.

Generally the best legal advice is to err on the side of caution. This is a creative industry, be creative.

1 hour ago, Michael Aganier said:

I read about trademarks that a same name can be reused if the field of usage is different enough.

Should be how it works. I remember way back when I made my company one of the requirements was that no other game development company had to have the same name. So a game developer can call them self Electronic Arts and a art production could name it'self Electronic Arts if they make art from electronics.

But laws change depending on where in the world you are. The details for naming should be on the forms you sign to register.