Sure. Why not? But let's try to clarify first what a portrait is.
Despite the implications of the accepted answer in the linked question, a portrait is not simply a picture of a person or a group of people. Or at least it shouldn't be.† The waters have been somewhat muddied by the shopping mall "portrait studios" that essentially do the same kind of job that a school yearbook photographer does, using fail-safe one-size-fits-all lighting, a mottled blue backdrop and a McDonalds-like manual of posing and processing to produce predictable results. That sort of approach may generate salable likenesses (there are no mistakes, which is more than can be said for most casual snapshots) but there isn't any portraiture going on. The same thing goes for models' and actors' headshots -- you want them to look their best, but you want to create a blank canvas upon which a casting director or stylist can build -- the last thing the client wants is a portrait. Commercial photography, at it's best, is a photographer's portrait of the model/actor's portrait of the character they're playing, not a "portrait" of the model.‡
So it's not about lighting, lenses, backdrops and poses. It's about creating a representation that is characteristic of the subject (or of the subject's public persona where the two are significantly different). A portrait should be both fingerprint and DNA -- a viewer should be able to look at the picture and say that it is of that one person and of no other. (The same goes for groups. There is a personality to each member of the group, dynamics between group members, and something characteristic of the group collectively.) If you take portraits of identical twins dressed and groomed identically, their friends and family should be able to tell them apart even if you can't after the fact. Portraiture is about assessing and capturing those things that make people who they are, not just getting the lighting, exposure and framing right. You need to be familiar with the subject or have the ability to read people quickly, and have the ability to coax the unique qualities out of the subject, in order to be more than a glorified yearbook photographer.
The same thing applies to pets. And yes, portraits of pets (and pedigreed livestock) have been around for a longer time than photography. Blame the English aristocracy and upper middle class -- they spent the greater part of the Seventeenth and Eighteenth centuries keeping artists busily employed with creating sometimes masterful pictures of wire-haired terriers, race horses and prize bulls. Admittedly, painters have an easier time of it -- they can imbue the subject with the personality and physical qualities that the owner imagines, while a photographer is more-or-less stuck with what's there. The pet's owner might see a characteristic head tilt or a "smile" whenever they look at their rather generic spaniel (or what have you), and in order to create a "portrait" of the animal, you've got to somehow coax the animal into looking the way the owner sees it. Oh, and make sure you catch the sparkle in their intelligent eyes (like the children of Lake Wobegon, all pets are above average).
The point is that you aren't merely trying to capture a likeness, you're trying to create an image that, to a viewer, could only be that one specific subject (and not the litter-mate that to you looks identical). The key thing to remember is that the root of "portrait" is "portray" (from an old French -- not Old French -- word meaning the same; the current French term, "dépeindre", bears a connotation closer to merely creating a likeness) -- if you are not capturing the essence of your subject, or at least what the owner thinks the essence of their precious pet is, you're just taking an "animal picture". People may pay for animal pictures, but they aren't going to be sending all of their friends and relatives to you if that's all they get.
† The "clinical portraits" mentioned are not portraits of the victims, but portraits of the disease or condition; as such they have a different aim. The idea is, again, not just to capture a representation but to characteristically represent the condition and its effect on the sufferer. That's pushing the term a bit, but if you remember what the pictures are aiming to capture, you can see that it's not about the afflicted person, but of the personified condition.
‡ Weddings are similar to commercial photography in some ways -- the whole day is a magical bit of dress-up and make-believe. Some real moments are very much worth capturing, but for the most part the photographer is there to tell the fairy tale rather than to cold-bloodedly document the day.