Sorry if this is a basic question, I didn't come up with a way to phrase a search to get any relevant results.

I am looking at replacing the crank on a 9 speed Trek 520. While I would ideally just buy a 9 speed branded crankset, I am curious if there is any reason one branded as an 8 or 10 speed wouldn't work. Is there an engineering difference between them that would cause me issues, or is more of marketing to sell groupsets as a whole?

The question has a second purpose as well, if an 8 speed crank with a 9 speed cassette is cool, the 9 speed cassette that is getting swapped out on the 520 for one with lower gearing is going on another bike.