Are you referring to a different sized mouthpiece for the saxophone? Or are you referring to the difference in size of brass mouthpieces?
Assuming you are referring to the saxophone, each type of saxophone will have a different size mouthpiece. By that I mean the soprano saxophone uses one size, alto saxophone larger, tenor saxophone larger than that, and baritone larger still. There are also some other rare models, but these are the main ones. Point is, mouthpiece size is largely dependent on what type of saxophone you're playing.
Now, I don't know what kind of saxophone you tried to play before, but let's say alto for simplicity's sake. Between all the different mouthpieces that will work on alto there are some slight size differences, but it's subtle and doesn't have a big impact. You may notice one piece has a slightly larger opening than another, but if you can play on one, you can typically play them all, especially with more experience. And this is true across all types of saxophones. I will say it is much more difficult to switch from playing an alto to a baritone, than it is to change the type of mouthpiece you use on your alto.
All of this being said, the real question is what do you with a music degree? Primarily, people choose performance or education. If performance, you need to focus on your primary instrument and complimentary instruments. I'm not sure what piano majors double on (it may be nothing), but saxophone majors often times will learn to also play the flute and clarinet. Reason being is often these parts are similar and composers will write them so that one person can perform a combination of these parts.
If you're interested in education then it is very helpful to be familiar with all of the instruments, and you will take several courses that introduce you to all of the different instruments you would find in concert bands and orchestras.