I dunno about “Platonic ideals”. I think there are particularities which would apply to any species that has individually sentient intelligent members and is also highly social. A hypothetical species in which the individual members are highly intelligent but the social aspect is a much looser affiliation (e.g., no language, most members tend to the acquisition of food on their own, minimal “herd” or “tribal” defense needs, very little in the way of endeavors that can be described as the outcome of intricate interplay of many individuals doing separate interwoven tasks the way that building a Mercedes-Benz is), would presumably be subject to different particularities. A hypothetical species that was social and perhaps intelligent in the collective sense (imagine an insectile hive with its own recorded history and myths of origin) but in which the individual members do not possess intelligence would, again, probably be subject to different axioms.
Meanwhile, there may be particularities that come from our evolutionary history as primates, things that would not be true of us had we developed from yeesh I dunno rodents or something. But the possibility of primates is itself the outcome of natural laws. That is not the same as saying there exists a natural law that will ensure that primates will develop anywhere that life develops, any more than life shall develop anywhere that a planet forms; but rather that there is a rich and intricate set of laws that govern possibilities for a planet with certain resources and in a certain temperature range and in which certain niches did appear and so forth, and structured within these is the possibility of primates. So the relevant moral particularities that pertain to the outcomes of certain behaviors as they may be specific to critters of primate ancestry are yet still the outcome of the working order of the universe as a whole.
Not all individually sentient socially interwoven species would necessarily come to embrace a set of axiomatic organizational principles that reflect certain laws about the most efficient way to be that very thing (individuall sentient socially interwoven yadda yadda)… but there are consequences for those that do and consequences for those that do not. Nowhere is it written in “Destiny stone” that our species is scheduled to persist for the next couple billion years. There are circumstances under which that is more likely and circumstances under which it is less likely.