I mean, I know the argument that gravity inverse square law becomes inverse cube law in 4d, but what I do not understand is that what/why enforces that. Why in a hypothetical 4d world there just can not be a gravity-like force that is inverse square? Would that cause some kind of contradiction?
This sort of intuition. Applies to electromagnetic waves, sound and gravity all alike.
This argument doesn't obviously apply to gravity (though presumably it would for a quantum theory of gravity), but the equations for gravity (general relativity) give the same result.
At a higher level, it turns out that when you try to combine quantum mechanics with special relativity, the resulting theories are highly constrained. It's not like classical mechanics, where you can just say 'suppose there's a 1/r^12 force.' You get mathematical inconsistencies if you stay too far. Weird stuff
It proves nothing of course. When we speak of these N+T universes, we try to imagine a system that follow the same "fundamental laws" but with different N and T. What exactly is fundamental is up to debate. You can even imagine a system that has different math, but it will be very hard to reason about it.