Watch Prime Video for two?

Hay.

My buddy and I have Prime and we'd like to watch a series together like Netflix does. And it should also be the case that when one pauses or fast-forward, it happens to the other in exactly the same way. Just like at Netflix Party.

Something like that on Amazon Prime?

As far as I know, Prime doesn't have a special addon like Netflix Party. What should work, however, is screen sharing via Discord (I haven't tested that with Prime yet, but it works with other providers). It's not quite as elegant as Netflix Party, since only one person can actually control the stream, but that's the closest thing to an alternative.