I seem to use distinctly different embouchures for tuba and euphonium. I think this shouldn't be altogether surprising since a tuba mouthpiece is significantly larger than a euphonium mouthpiece and there is more room to "move around in it" for different effects and results. I really noticed this difference when I was playing the euph this year for all the holiday music and events.
My "default" tuba embouchure seems to be something along the lines of medium-low to low, shifting more in the low direction for the the contra bass area. I also will tend to get a bit more of the lips into it for very low range as well. Keep in mind that I don't use one of the enormous tuba mouthpieces that have become fashionable in the CC and BBb world (like the PT-88). I regard the Wick 2XL as slightly big for me. At 32mm rim diameter and a rounded inner rim, it's right at my limit for that. And it's pretty deep. The 3XL is actually a bit more comfortable, but I just can't get decently rich tone with it. That said, my upper range with the 2XL is fine, and is very close to my upper range on the euph . The 2XL requires more focus and control on my part than does the 3XL (or a Schilke 66, or TU17, or even PT63). In the upper range, my embouchure becomes closer to my euph embouchure.
My euph embouchure needs to be a bit more precise overall, and tends to be more in the genuine "medium" (or "center") range. Since I use a fairly large euph mouthpiece, I don't do much, if any, shifting for the low register -- or so it seems to me.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)