Well, the mouthpiece doesnt care what its branded as, but a small chamber large shelf baffle piece will sound a lot different to a large chambered piece with no baffle, so yes, different mouthpiece designs will have a big influence over the type of sound you can generate... If you get two brands that are of similar design, you might not be able to hear a lot of difference.
Thats not to say that a great player cant make whatever sound they want too within limits, regardless of the piece they are using, but for us mortals, the right mouthpiece can be a usefull shortcut to a particular sound concept.