The most significant genetic feature of the human species is the brain evolving the capacity of speech and the ability to learn and teach new things.
This human reliance on speech, whence clothing and jewelry and their symbolic uniforms to artificially create new self-identities, is so radical, many humans cease to even self-identify with other species of animals. Even bonobo and chimpanzee who are genetically similar to us can seem unrelated.
Certainly our intelligence sets us apart, and so does speech (though it seems we are learning more and more that other animals are capable of pretty complex communication). Still I can imagine a world where bonobos, chimpanzees or gorillas evolve alongside humans into equally intelligent beings capable of speech like humans, but whose physiology gives them different cultural tendencies. I would imagine if a chimpanzee retained its strength (which is significantly greater than a person's), that would shape their culture. And I can imagine a world where neanderthals remain, and their culture might be substantially different from human culture. We know Neanderthals made music for example. I would be curious if their hearing senses were similar or different from our own, and if there is a difference, how different their musical tendencies might be. I'm not informed enough about other primates to continue this thought experiment without further research, but you get my point.
And I think that is another thing that is significant: these are thought experiments. They are based on things that don't exist. We live in a world where humans are the only advanced and intelligent species (there are other intelligent species but you get my point I think). So we have to imagine what other species of hominids might be like, or other beings who might vaguely resemble humans but have different physical features, builds, life spans, etc. You and I might answer that thought experiment differently, and it seems we do. But I also think there is room for legitimately different answers here.
Speech is also why culture matters more than instinct. Different human groups learn and teach different ways of adapting, surviving, and flourishing. These artificial differences are of more consequence than any trivial genetic difference. Humans evolve by means of evolving cultures far more rapidly and dramatically than by means of mutating genetics.
I agree with you our speech is very important, and I agree we have a variety of cultures, a variety of ways of adapting. But I think you could also flip that and say our intelligence, our ability to adapt, is one of the things that shape human culture (again we are doing a thought exeperiment because we don't have other cultures to compare to). But I would push back against the role of instinct, the role of our biology. I think we like to think of ourselves as not part of the natural world, but we are. And we have drives that shape us, and affect our culture. We might not have a predators reflexes, but we are not purely cultural beings. Obviously things vary, and culture can stretch what is possible, but there are basic human relationships (like immediate family) that are hard for culture to uproot. We also need things like human interaction, we need to laugh, we have, I would say, a pretty clear inbuilt instinct to make music and perform rituals. Those can be expressed in different ways in specific cultures, but there is an underlying humanity there that I think is part of our nature.