Alternative theory: "humans are pretty smart and good at understanding animals." There's a fair amount of supporting evidence for that theory. I can understand a dog's gestures pretty well too. Does this mean I have some kind of Dune style genetic memory of when my ancestors first tamed wolves?
We know that "genetic memory" is a thing because there are experiments demonstrating it. For example, human (and, more generally, primate) babies seem to have hardwired ability to recognize snake-like shapes as threatening.
It doesn't work like Dune, of course. It's just the consequence of evolution applying to the brain just as much as it does to everything else, producing brain structures that are dedicated to a particular task that happened to be important for the species at some point. If pattern-matching snakes is such a task, I don't see why pattern-matching primate gestures couldn't be one, as well.
I also find myself intuitively in the camp of your alternative theory.
Nevertheless I have an off the cuff comment to your question: Dune genetic memory is outside the bounds of our most basic understanding of inheritance because we do not know of any obvious mechanism to encode those memories into gamets in "one shot" (from direct experience to immediate offspring). Compare this with a very long arc over which a gazelle is born with brain initialized so carefully that it can learn to walk instantly, effectively having something that really looks like "genetic memory" of how to walk.
My confused ramblings aside, this to me suggests that it is at least plausible that such commonality among apes and homo exists. Now, providing evidence for it is quite a different ball game ...
It seems like a pretty basic component of anything resembling intelligence. Even relatively simple animals tend to be be able to demonstrate some form of communication with each other. This animal is showing its teeth to me, the teeth it would use to attack me, posturing itself in an aggressive way, and making some noise. Ah I guess that noise doesn't mean let's be friends.
So I think your question is where the intelligence came from, and outside of hand-wavy appeals to evolution, I think the simple answer is that nobody knows.
Remains to be seen, although it's noteworthy that when around cats, dogs, horses, or cows; humans will pick up on their body language very easily, even though we have relatively distant common ancestors.
