Tags need to be distinct or they're not useful. The only distinction that makes animal useful to have on the site is that it labels something particular that's different from what monsters labels. To do that it has to be distinct in how it's used.
The animal tag's history of use suggests that the community thinks that the distinct use for animal is to label questions about normal, mundane creatures — the kind of creatures you might find in a farmyard, as a beast of burden, or in a hunting forest on Earth.
So let's look at how the tag has been used. (In the process I found out that we had a pile of them that should have been tagged animal-companion instead, and fixed that in the meantime. This doesn't undermine my point because, at worse, it removes questions that would have supported my point.) The questions appear to fall into five categories:
Domestic animals and pets (5 of 23, 22%)
Wild mundane animals (8 of 23, 35%):
Unspecified animals, but mundane from context (6 of 23, 26%):
Giant versions of mundane animals (3 of 23, 13%):
Monsters that are only superficially like a mundane animal (1 of 23, 4%)
That makes for 83% (19) questions being about very normal, mundane animals, and 17% (four) being about animal-like monsters. Sliced another way, that makes for 97% (22) questions about Earth-appearing animals of normal or giant size and 4% (one) question about a creature that doesn't appear on Earth in any size.
The question about the shadow mastiff is an obvious outlier, and is probably mistagged — it's probably more sensible to label it with monsters instead of animal.
The fire beetle is
similarly out of place: it's not a mundane animal, either domestic, wild, or giant. It's a D&D creature that's was invented and is only superficially like a mundane animal. In this breakdown it would be in company with the shadow mastiff; one question is the weakest of precedents. an oddity, in that it is and has been a D&D “monster” for decades without the connection to a real-world animal being obvious, but it's apparently (whether by design or accident) very similar to a giant luminous click beetle.
So given this look at how the animal tag is being used, a clear distinction is being made between monsters and animals. That's good, because having a clear distinction is the only reason the tag is meaningful and useful enough to keep around, instead of merging it into monsters. To maintain that distinction, questions about monsters should be kept out of the tag.
There is probably room for, as doppelgreener suggests (and which I already upvoted), a few odd creatures to have their questions categorised by both monsters and animal, since there are examples that are on the borderline between them and not obviously exclusively either/or, such as the dire toad and the fire beetle (and possibly giant chameleons used as riding mounts…).
The point of putting monsters on a question, regardless of what the wiki says, is so that when someone is looking for questions about monsters, they can find questions about monsters. Since giant toads, fire beetles, and such are widely considered monsters by the playerbase of D&D, they should turn up when browsing D&D questions tagged with monsters. Similarly, creatures that are archetypically animals should show up when browsing animal; I don't think fire beetles (or dire toads or giant chameleons) are what people browsing that tag will be looking for, given its overwhelming use for very mundane, domestic animals… but they're on the borderline, yes, and an argument can be reasonably made for including them.
So to answer the titular question: yes, the tag wikis don't seem to describe how the tags are actually used, so they should be amended to match actual usage.