But, here is my question. Does MWI limit itself to alternate universes with the same universal constants, or does it predict also the existence of universes with different universal constants?

As far as I know, we don't know why we have the physics constants we have now. There are hints that the constants may be a product of the structure of the universe (and that the constants have changed over time as the structure of the universe has developed), in which case MWI would predict universes with different constants. But there are a lot of unknown unknowns in all of this.

Insofar as MWI might predict universes with completely different physical laws - I'm unaware of any evidence for this proposition.