Why is this any different?

The judgment in the article I linked goes into detail, but essentially you're asking for the law to let you control something that has never been yours to control before.

If an AI generates something that does indeed provably contain a sample of a piece of music in a song you recorded, then yes, that output may be something you can challenge as a copyright violation. But if the AI's output doesn't contain an identifiable sample, then no, it's not yours. That's how copyright works, it's about the actual tangible expression.

It's not about the analysis if copyrighted works, which is what AI training is doing. That's never been something that copyright holders have any say over.