I broadly agree with Valorum's answer, and his position on the overall scope of this Stack. In other words, I believe that merely having an anthropomorphic animal is rarely sufficient to make a story on-topic, and I mostly agree with Valorum about what the exceptions to this rule tend to look like.
However, I don't entirely agree with the reasoning that Valorum uses, so I'm going to explain how I came to that conclusion. Think of this as a concurring opinion, if you like.
TL;DR: Skip to the bold sentence and read from there to the end.
There are a wide variety of stories involving anthropomorphic animals. It is difficult to generalize over all of them, so I'm not going to try. Instead, I'm going to look at a few specific cases and try to divine a general-purpose rule.
Disney has had a rather significant level of involvement in mainstream talking-animal stories. While Disney's stories are not at all representative of other material, they are popular and likely to come up frequently in these discussions. Much Disney work involves adapting traditional folk tales into children's movies, and this often includes replacing the human characters with animal characters (e.g. Robin Hood, 1973). Other Disney stories are not directly adapted from folk tales, but still use animals as stand-ins for humans (e.g. everything involving Mickey Mouse and company). Other animators have done this as well; Disney is only getting name-dropped because they're popular and easily recognized.
I think we can all agree that most of the material I just described is off-topic. Sure, there have been some Mickey Mouse stories which involved magic or other obviously fantastical elements, but we're not going to admit every Disney story with talking animals. Merely replacing humans with animals is not sufficient to bring a story under the umbrella of speculative fiction,* if it wasn't already there to begin with. So, our first approximation is this: perform the reverse transformation, replacing animals with humans, and then ask whether 1) the resulting story would be on-topic in its own right or 2) the resulting story does not make sense. We need condition (2) to rule out stories in which the characters really are animals, and not just humans who've been written as animals.
But we can't stop there, because this first approximation doesn't always produce correct results.
One obvious example is Bambi (1942). In this story, the title character's mother is killed (spoiler alert) by a hunter because she is a deer. If she were human, the story wouldn't have made much sense. But I for one don't think Bambi is on-topic just because of this. Perhaps if you're going to replace the animals with humans, you also have to replace the humans with story-appropriate plot devices. Under this transformation, the hunter who killed her is replaced with some natural disaster or other calamity. The resulting story is entirely mundane, and clearly off-topic. Similar examples can be found in the Pixar canon (see particularly Finding Nemo, 2003) and elsewhere.
This is, however, not much of a standard, because if you allow arbitrary replacement, you can transform any story into any other story. You can replace werewolves with humans, lycanthropy with some real disease, and so on, and eventually arrive at a mundane story. Werewolves are on-topic, so this approach must be flawed.
Instead, I think it helps to examine the narrative focus of the story:
- If the humans treat the animal like another human or like an animal, or both at different times, but the narrative does not really ask any interesting questions about that character's animal side (e.g. Scooby Doo), then the animal probably does not make the story speculative fiction (but it might still be SF by virtue of some other story element, as with the Scooby Doo movies that have real monsters and such).
- If the humans treat the animal like another human, and this aspect of their relationship is explored extensively, it probably qualifies as speculative fiction. For example, The Elder Scrolls can be considered fantasy solely on the basis of the Argonians and Khajiit, because there is a lot of backstory involving political and military interactions between the Empire (in various incarnations), Black Marsh, and Elsweyr. It's already fantasy six ways to Sunday, though, so this may not be the best example.
- If the humans are nonexistent, the story might still be speculative fiction, if it focuses on (for example) the interactions of different species with one another in a highly counterfactual setting (Zootopia, 2016), or if the main characters are aliens or fantastical beasts, or otherwise "obviously" speculative.
Our final criterion, then, is this:
If the animal's non-human nature has substantial narrative focus, then the story is probably on-topic. Stories that pass this criterion tend to ask and answer questions such as the following (not an exhaustive list):
- What would it be like to be an intelligent [animal]?
- How would a mixed society of humans and [animal]s work?
- How would a mixed society of [animal]s and [other animal]s work?
- What would an intelligent [animal] think of humans?
- What would a human think of an intelligent [animal]?
- Why are these [animal]s intelligent, anyway?
Stories that ignore or avoid these sorts of questions probably fail this criterion. Stories that occasionally touch on these questions for character development purposes, but otherwise ignore them (especially for plot development and broader world building purposes), are likewise probably outside this criterion.
These questions are meant as illustrative examples, and should not be used as a checklist. There may be off-topic stories that do touch on one or two of these questions, and there may be on-topic stories that hit different questions, or even reach topicality by some route other than anthropomorphism (e.g. Fantasia - the wizard's hat is fantastical, but Mickey Mouse isn't). Topicality needs to be examined carefully in each case. The best way to determine whether something is on-topic is to ask a meta question and tag it with discussion and scope.
* "Speculative fiction" roughly means "science fiction, fantasy, and the stuff in between."