I think, and certainly in my experience, there is a world of difference between the saxes when it comes to embouchure. For me, playing alto pretty well (from a sound/intonation perspective at least), it was a real shock moving to tenor. It seemed everything was harder; more resistance, more breath support needed, totally different embouchure - much more relaxed at the bottom end but less so in the upper register. It took me a fair few months to make the transition from one to the other.
Then I go buy a sop! Moving between tenor and sop was such a nightmare for me that I haven't used the sop in the last 8 or 9 months at all. My teacher advised I shelve it as it was destroying my nice laid back and relaxed tone on tenor. In fairness, I just have the original and another cheap plastic mouthpiece for the sop and I never did any trials whereas for tenor I spent weeks and tried out maybe a dozen.
I just have the one mouthpiece on tenor, a Jody Jazz DV NY. I've been playing sax just short of 3 years and tenor for about 21 months. Embouchure takes a good few years to develop and I still feel mine is still improving - less so in the 'best' quality of sound and more so in my ability to produce the same consistent sound every every time I pick up the horn.
What has also helped IMO is sticking with just one mouthpiece and in my choice of reed. After almost 2 years using Marca Jazz 3s I just a couple of months ago switched to a synthetic
Légère Signature which is totally brilliant. Initially I thought the 3 was a little hard but having tried the 2.75 I am now convinced the 3 is the right one for me. It plays from cold exactly the same time after time.