When this baby boomer was younger, we always said, "he plays the piano." More and more I hear and read, "he plays piano." Is there any guidance on this?

I use "I play the trumpet" more frequently than "I play trumpet", but find both acceptable.

I've heard "I play piano," though less frequently than "the piano." I haven't known that many people who play the trumpet, but I've never heard "play trumpet."

I have usually heard the article omitted when reference is being made to a part in an orchestra or similar.

"He plays 1st violin in the LSO"

I heard the shorter terms quite frequently in school. It doesn't necessarily make them correct, but you know how we love to speed up the communication. I wouldn't be surprised to hear, "I piano", "I tuba", or "I trumpet" now.

I don't know that piano or trumpet can be an adverb, but perhaps it is an adjective with something like music being the implied subject?

I play piano. = I play piano (music).