<button> already behaves differently depending on the type of form it's in, or whether it's in a form at all.

Let me put it another way: If you use a <select> today, what's the difference between the thing that renders on the page and a <button>? They look different, sure, but that's a matter of styling. You can already style a <button> to look like a <select> activator.

They can receive focus, and they can be activated. They're buttons.