** I am not a lawyer - nor an IP Expert just a person with some Local knowledge so take with a grain of salt **
DBA - Doing Business As it usually a local County registration.
LLC is state level and can be recognized outside the state
C and & S Corps are also done at the state level and then file for operation in other states.
Some people incorporate ( not usually martial artist as the volume of cash flow may not be there ) in special states that make it easier for tax reasons.
If the name is in the same country I would worry ( a little ).
If in the same state, you might be able to have a different form e.g. XYZ LLC versus XYZ Inc.
Check out your local state government for these types of entities.
Reach out to a lawyer if you have any serious questions.