Interesting. Differences, but there is certainly still a common core...

[A mad genius attempting to take over the world with his machines. Curious that the more successful one is the one who also has (or at least had) a dependable human ally. Did a parallel Stone not exist in the other universe? Or maybe one did and something happened in the past, something similar to what may have happened in this world now, and that is why the other Robotnik typically only trusts machines. Perhaps this timeline is only now reaching that event, and that conclusion?

Are the two universes even similar enough to make such hypotheses reasonable? Is the multiverse truly infinite, leading to all possibilities existing simultaneously, or is there some basic template that leads certain things to always be more or less the same? The latter makes it far more statistically probable for Infinite's world to be so similar, but that's not evidence; it's likely that the Doctor sought that out deliberately. Could you find a version of Dr. Ivo Robotnik who was - he doesn't know, a kindly toymaker or something?]

Are Sonic and the others his enemies there as well?