“Beethoven .. ” This video is no longer available due to a copyright claim

(I assume, without knowing the specific example) they are not claiming copyright on the piece, they are claiming copyright on a specific recording they created.

I once put up a YouTube video of "Auld Lang Syne." I was playing this song myself on guitar, double-tracked to provide a backing track. I got a copyright claim from YouTube.

The lyric we all know was written in 1788, although he may have adapted existing folk lyrics. The actual melody is probably borrowed from an older folk melody. And of course since it was my performance, I held the copyright to it.

In this case as others have said, the copyright is on the recording. But YouTube's automated copyright-infringement detection is legendary at being somewhat dumb.

It's their performance of the music that is copyright, not the music itself.