Let me try to be clear about this so there is no misunderstanding.
Natural rights doctrine notes that we -and we alone - have these rights due to our humanity, the implication being that ‘humanity’ is a distinct category - i.e. that we exhibit a fundamental difference in kind from all other creatures.
So obviously, the question of what exactly is the fundamental distinction between humanity and other creatures presents itself. What is is that is ‘human’? Is this simply due to the taxonomic nomenclature created by Carl Linnaeus, which separates humans into our own genus (Homo) distinct from the other primates??? Obviously not, especially since Linnaeus himself noted that the only reason he thus separated humans out from other primates was out of fear of running afoul of the religious authorities. TH Huxley, among others, made compelling arguments that humans belong in the same genus as primates, and should be separated out at the species boundary only.
So what does this mean for the natural rights arg? We seem to assume that the thing that makes us distinctly human - the difference in KIND - is the thing that results in our having natural rights. But based on the scientific evidence, there is no ready answer to ‘what is the unique differentiator which expresses this difference in kind?’ Comparing what we now know about chimps, especially, blurs this once-seemingly-clear boundary.
For example, many philosophers - from Plato to Hegel - have held that we are fundamentally distinct due to reason, intellect, understanding, etc. But there is overwhelming evidence now that many animals reason to some degree or other, particularly primates. So this is a difference in degree, which in fact was the view of both David Hume and Darwin.
We used to think self-awareness was the differentiator. Turns out, this is wrong - chimps are also self aware. Again, difference in degree.
Then we thought - ability to communicate via a true language was the differentiator. Turns out chimps can learn a substantial amount of Amslan and use it effectively. About as much as a 2 year old child - we’re not talking real complex communication, but linguists have been astounded that chimps could accomplish as much as they did. So we can’t discount them on that criteria - it’s again a difference in degree, not kind.
Social organization and culture - same issue. Differences in degree.
Mortimer Adler encapsulated the latest philsophical thinking by laying out 4 definitive criteria (see his The Difference of Manand the Difference it Makes for more) - and each of these ‘confident assertions’ has since been destroyed by scientific advances in understanding chimps, especially.
So - if we have these rights by virtue of our humanity, and if we reject the ‘Creator-given’ and ‘soul’-based arguments (which I do), but we can’t really come up with a definition of ‘human’ that excludes chimps (aside from a DNA argument - but who claims natural rights spring from DNA???), what makes us so damned sure that chimps are not also deserving of these rights?