It's clear that we have a linguistic gap created by the fact that "animal" has two meanings -- members of Kingdom Animalia (which includes humans), or all members of Animalia *except* humans. This linguistic gap can create confusion in conversation about animal rights, and lead to people making false inferences from the fact that (someone says that) humans are animals by the first definition to the conclusion that (they are claiming that) we have the characteristics of animals by the second definition. I'm not sold on solving this problem by introducing "anymal" for the second definition -- it seems too close in spelling and pronunciation to "animal." But this is in a general sense a reasonable thing to try to do.
The use of "anymal" as a position-signaling shibboleth and as a conversation-starter I'll leave to the side for now.
Kemmerer's main argument is that our mistreatment of "anymals" derives from a dualistic hierarchical conception, and that the use of "anymal" can challenge that conception. I agree with the former, but not the latter. Beyond its signaling and conversation-starter qualities (such that using the term is like adding a footnote to everything you say that says "remember not to think dualistically!"), I don't see how the term itself directly challenges dualism. The underlying concept is the same, and is equally dualistic, whether we label non-human animals "animals" or "anymals." Kemmerer quotes J. Dunayer commenting on the absurdity of dividing Kingdom Animalia into squids and non-squids. This is only absurd because it's hard to think of a situation in which we would need to refer to all non-squid animals together -- but if such a linguistic need were to arise, it would be perfectly sensible to refer to them as "non-squids." And in fact we do have a linguistic need to refer to all non-human animals as a group, since prevailing ways of thinking treat them all as less morally worthy on the basis of their non-human-ness.
Kemmerer claims that the alternative term "non-human animal" reinforces the dualism by defining one side by its not-us-ness, and instead we need a term that puts both sides on an equal footing (there are parallels here, though she doesn't explicitly draw them, to the use of "person of color" instead of "non-white person," and in a more complicated way to "cis" instead of "non-trans"). I'm not entirely convinced on this point -- after all, the only thing that bees, bears, sea cucumbers, and sea gulls have in common with each other but not with humans is that they have historically been defined as lower on the moral hierarchy due to their lack of humanness and treated accordingly. But even if we accept that part of the argument, "anymal" doesn't solve it. Kemmerer defines "anymal" as a contraction of "any animal who does not happen to be the species that I am." Thus the not-us definition is right there in the etymology, albeit pushed back a bit out of view. If we can't even define the term without using dualistic thinking (contrast our ability to define "people of color" or "cis" without contrasting them with white people and trans people), it's going to be a dualistic term no matter what set of letters we use for the label. Kemmerer is certainly right to argue that our ways of talking about things can shape how we think about them and can even amount to a form of activism -- in the animal rights context, for example, we should oppose expressions that say that misbehaving people were acting like animals. I just disagree that the "anymal" terminological change fits that bill.
Kemmerer tries to draw parallels with replacing "chairman" with "chairperson" and discarding terms like "Negro" or "cripple." With respect to the chairs, the problem with "chairman" is that it puts the presumption of male leadership right there in the term. No such presumption of animal inferiority is in the term "animal" or "non-human animal." With respect to disparaging terms, "animal" is actually quite morally neutral in its usage -- it is neither inherently disparaging (like "cripple") or indissolubly connected to a history of oppression (like "Negro"). This concern could apply to a term like "brute," though I note that the only people today who seem to use "brute" in its original literal sense as applying to non-human animals are animal rights activists criticizing the word.
In sum, while it would be helpful for conversational clarity to have different terms for the two meanings of "animal" depending on whether you're including humans, I doubt I'll be using "anymal" anytime soon -- and whatever term I use, I won't expect the term itself to overturn dualistic hierarchical attitudes that enable mistreatment of non-human animals.
*The specific article, "Killing Traditions" from Ethics, Place, and Environment in 2004, was a strongly pro-animal-rights and anti-tribal-sovereignty take on the Makah whaling controversy. On this particular issue, I'm much more in sympathy with Greta Gaard's 2001 article in Hypatia "Tools for a cross-cultural feminist ethics." Were I to be asked by the tribe for advice on whaling, I would reply that killing whales for reasons other than physical subsistence or obligatory environmental management** is wrong. However, in light of the history of dispossession and assaults on their identity and livelihood faced by Native Americans, I am extremely reluctant to advocate the abrogation of treaty rights if the Makah decide they want to hunt. Further, as an outsider I have neither the right (due to general considerations of cultural autonomy and specific considerations of the historical and ongoing power imbalance between white and Native Americans) nor the ability (due to a lack of knowledge about traditional and modern Makah culture and a lack of acceptance by the tribe as a legitimate interpreter and re-performer of that culture) to propose how to integrate non-killing of animals into their socio-ecological context. That's the job of anti-whaling and animal-rights activists from within the tribe, of which there are several. Indeed, it may be that the best thing an outsider can do for the whales is the best thing for the Makah as well -- strongly defend tribal sovereignty and oppose other outsiders' racism and paternalistic judging of what's genuinely traditional, which can give the Makah more breathing room to maintain and rework their culture for modern circumstances, a reworking which may -- but can't be forced by outsiders to -- include restructuring their relationship to whales and other animals.
**Some in the pro-whaling camp -- and I'm not enough of a marine biologist to know whether this has any truth to it -- argue that 20th century restrictions on whaling have led gray whales to become overpopulated. Culling in such circumstances is a complex question, but is not something that I think can be prima facie ruled out.