Is there a similar strong will or intention in how a multiverse evolves?

Well, if we're talking about the many worlds theorem, then probably yeah, because both worlds came from a common starting point and evolve together. Like, imagine that I flip 100 quantum coins, creating 2^100 (1,267,650,600,228,229,401,496,703,205,376) universes in a multiverse. Every universe will be different, but the vast majority of them will have roughly 50 heads and roughly 50 tails. 7% of them will even have exactly 50 heads. There is one universe where every coin flip lands on heads, but it's only one universe among nonillions, you could spend your entire life searching universes and never find it. None of the universes are the same, but most of them are so boringly similar that you couldn't tell them apart. It's the central limit theorem, that lots of random events trend towards uniformity

nobody really knows, but if I had to guess I'd say that's probably the way our universe would be, our universe might technically be different from the one next to it, but it would only be different by a single electron on mars that decided to move an atom to the left. There might be a universe somewhere where all of the particles in a lotto wheel quantum tunnel to make the winning number be your number, but it would be outnumbered an infinity to one by universes where that didn't happen and it looks exactly the same as ours.