What constitutes a professional musician? Is it skill level, or is it when they're paid to play music? I've always heard that it's when someone is paid to play. My brother thinks it's skill level. Which of us is right?

Most answers already told you that you are correct; especially in popular genres (by definition, popular music can be enjoyed and performed by people with little or no musical training).

What makes the definitions slightly confusing is that you can’t become a professional in genres like classical music or jazz without having a certain set… sorry, level of skills. More recently, even the skills aren’t enough in many places without an academic degree.

On the other hand, skills or degrees aren’t enough to make you a professional musician: if you don’t work in the domain, you’ll be considered just an amateur.