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Of course there are lots of tubas of any key (F, Eb, CC, BBb) any size (3/4, 4/4, 5/4, 6/4) and any style
(German, -->somewhere in the middle, -->American).

But let's say that on any tuba you happen pick up and play; what cup size, cup shape, back bore size, (and in what combination) would you expect to create a sound that is:

1.) dark/mellow
2.) dark/edgy
3.) bright/mellow
4.) bright/edgy
5.) broader
6.) more focused
7.) horn plays too flat overall (main slide is too short to cut)
8.) horn plays too sharp overall (tuning slide is too short)

Any catagories I missed?

I'm really interested in what direction(s) this post could go...

If we leave your face out of the equation, then more cup volume and a bigger throat (within reason) usually result in a bigger, deeper sound. I'm sure there are exceptions to this, but I'll go ahead and say that the harmonic content and focus is determined by what your lips are doing. However (aha!) what your lips do will almost certainly be greatly influenced by the width and shape of the rim and the MP diameter. And that is where you get so much variation in YMMV.

With that said, grabbing a bigger MP won't necessarily get you a bigger, more robust tone if your embouchure can't make full use of the bigger MP. That is: If you play a 33.5mm MP with a 32mm embouchure, you are not going to get a robust, well focused 33.5mm tone.

I haven't yet come to a decision what I think about claims of mouthpiece to horn match. In general, I just haven't observed that. EXCEPT, my recent acquisition of a PT44 and use with the 191 has been getting me "the hand" (or verbal equivalent) when I thought I was keeping it down. My inclination is to conclude that my brain and face happen to naturally create an extra resonant tone with the PT44 rather than there being some extra special compatibility between the 191 and the PT44. But that is pure speculation.

That's why you hear, over and over, you have to play it for yourself to know what it is going to do.