Do you mean the name of your song or songs? Do you mean Paris Davidson, Jr.? Do you mean your band or DJ/Producer/Songwriter name, or your business entity???
These are all different, and it's hard to answer with the vague wording of your question...
In brief-- The names of songs are not generally "copyrightable." The music (harmony and melody) and lyrics are, as is the individual recording/master recording itself. But not the name (which is why there are so many songs with the same or very similar sounding names).
As to your business name, it depends on which you're talking about. There are corporations, incorporations, LLC's, and DBA's. As to federal and international copyright protection for names, logos, and the like, they are not submitted for copyright, but rather for trademark
or servicemark (tm or sm).
As to your given name, if someone else legally has the same or similar, or wants to legally change their name to Paris Davidson, Jr., they may do so, and do business in the same way you do. But there are possible protections, if you feel (and more importantly can prove) that they are impinging on or negatively impacting your business in some way. This would involve a lawsuit and charges of "restraint of trade," "unfair competition," or maybe even slander or libel or defamation, depending on the situation.
It all gets pretty tricky, and for clarification and disclaimitory purposes, I will state that I am not an attorney, nor do I play one on TV. Seek competent legal advice.
So, what did you mean when you asked the question???
GJ