|Obligatory Venn diagram|
Music is your main thing.
You don't have a full-time career outside of music. There is some wiggle room here: a lot of people— maybe even most, now, times being what they are— have to supplement. Some great musicians I can name took up other jobs at some point in their career; they didn't stop being professionals just because they gave up performing for a job as an accountant after 35 years in the business.
You have been paid for it, and are oriented around getting paid for it.
Traditionally in music the bar for admission is very low: you're a professional after you've had your first paying gig. The other things here have to be true, as well, though. Your long-term orientation is that your job is musician, even if you're not very successful at it right now.
You have professional training and expertise.
You studied music in college, or are otherwise educated in music, you have field experience, you read the professional literature.
You are able to do professional jobs in your local scene.
You may or may not be able to give Thomas Lang a run for is money, but you are able to do most jobs in your city, usually with little or no preparation or rehearsal.
You understand and follow professional practices.
You know how the business works at your own level, at least. You get yourself to the gig, on time, properly attired, with everything you need for the job, and you perform the job capably, with a professional demeanor. You know how and how much you are supposed to get paid, you know how to do your taxes.
You are a member of trade organizations
This one is sort of optional, as a lot of people are not members of any organizations whatever. But: you have been a member of the local Musician's Union for some period of time— with the nature of the business today, most club musicians I know are not active members their entire careers, however. You are a member of other trade organizations like PAS, MTNA, NARAS, etc. You attend trade events like PASIC, NAMM, IAJE conferences, etc.
You maintain a professional studio.
You have a workspace for doing all your non-performance musical work: practicing, rehearsing, teaching, writing, and office work. You have a collection of instruments adequate for the work you do. You have a computer with whatever software and peripherals you need for your music business. You have a library of recorded music, drumming books, and general music books. You have archives: recordings, published works, recorded masters, other written works. You have a certain amount of AV equipment: stereo, audio and possibly video recording capacity.
Professional does not mean successful
Vincent Van Gogh was a painter who happened to be extremely unsuccessful during his lifetime— his business was a total failure, to the point that it killed him and his agent, his brother Theo. But Vincent was a professional: he did nothing but paint, he had professional training, he was recognized as a peer by other professionals, and was represented by an agent for the purpose of selling his paintings. He was a professional, but he never sold a painting. And the fact that his pictures now are valued in the hundreds of millions is irrelevant, actually— he's just a convenient example.