A fallacy is a bad argument. In order to determine a type of fallacy, one has to present an argument. I'm not entirely clear what argument you are making. It helps if you state it as a one or more premises followed by a conclusion. It seems like what you have in mind generally is:
P1 In a collection of species, some species are more similar than others.
P2 If a species is radically different than the rest, then it is somehow unique.
C Therefore, a unique species is misclassified and its similarities are coincidental.
In the case of your post, it seems to me the argument made is:
P1 All evolutionary species are distinct, but related.
P2 Human beings are radically different in terms of evolutionary criteria because of things like language use.
C Therefore, being unique in the use of language, human beings are not a product of evolution but only appear subject to evolutionary forces by coincidence.
I'm not sure there is a fallacy here. Remember, people misclassify things regularly. If human beings do not have sufficient similarity to the rest of the categories of evolution, then it's reasonable to conclude they are not a product of evolution. The form of the argument is not the problem. The problem is with the second premise which suffers from a certain bias, some combination of egocentric bias and specieism: humans are NOT fundamentally more unique in terms of evolution, even if language has made us the dominant biological organism on the planet. The end result is either an unsound argument formally or an uncogent argument informally.
So, you can have a flawed structure of argument or you can have a flawed premise, both of which lead to a flawed conclusion. In this case, the second premise is fatally flawed. How?
First, start with the fact that humans, bonobos, and chimps share almost 100% of their DNA (98.8% according to this article (amnh.org)). If you understand what genes are, to claim that we are relatively unique is a strange claim. In fact, one famous primatologist, Frans de Waal has written a fantastic string of books showing not only are our genes almost identical, but a lot of human behavior is relatively similar to chimp behavior. Proto-language, proto-morality, and politics are all present in chimps and bonobos. The big difference between us and them seems to be that we are capable of grammars, and they are not. Michael Tomasello (GB) has a series of books that makes an overwhelming philosophical case of the similiarities from the perspective of communication, cognition, and collective intentionality (SEP).
Why is that so tough to see for some people? Education systems in religious countries sometimes forbid or undermine the knowledge, as here in the US or attempt to conflate it with creationism. Quite frankly, it requires a certain sophistication to understand linguistics, evolution, and philosophy necessary to comprehend the argument. Grammar, the genome, and metaphysics aren't topics that most people ever develop much of an understanding of. Also, there are psychological and sociological incentives such as egoism and in-group-out-group incentives to deny even other human beings full in-group status. People routinely commit murder and genocide to get what they want. Peter Singer routinely makes arguments that human beings should honor their altruistic heritage and use degree of consciousness as a criterion for treating other animals better (such as stop killing them with hydraulic presses and eating them), and there's a lot of resistance to that thinking. As they say, it's hard for someone to accept a truth when their paycheck (or meal) depends on thinking otherwise.
As for what fallacy is possibly applicable in this argument, off the top of my head, I'd consider special pleading:
Special pleading is an informal fallacy wherein one cites something as an exception to a general or universal principle, without justifying the special exception... It is the application of a double standard.
Human beings are cited as an exception to the principle of evolution which isn't an attack on evolution, but merely an attack on our conclusion thus opening the door to questions about alternative origins of the human species. This sort of argument feels like one made of a Christian natural theologian who wants to preserve both evolutionary theory and man's special place in the Divine Chain of Being.