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Actually, I believe it would be a guild, as artists are not actually employees of any particular label or service, but just contractors. I also thing Guild is a far cooler name, but that's beside the point!

As for your actual question, no I don't think they are. They seem more like an advocacy or lobby group

https://www.unionofmusicians.org/#about

Well they ripped off the (copyrighted) Mr. Yuck logo owned by the UPMC Children's Hospital of Pittsburgh.

Stealing from children to complain about theft.

It looks like yet another cosplay of 20th century labor politics, but with the typical 21st century twist of diverting the fight from those who wield power to... the paying customers, for whatever reason.