It's not a Targ. The targ was invented afterwards and looks nothing like this animal, nor it has any reason to.
By this logic, pugs aren't dogs because they don't look like German Shepherds.
The animal in III doesn't look like a targ, doesn't walk like a targ, doesn't quack like a targ. The only thing it has in common with a targ is that both appeared in Star Trek, and one of them only from TNG on.
If Kirk found it in the ship after the movie he probably barbecued it since there is no meat to eat in Vulcan.
Now, pugs? what pugs? and you talk about my logic?
I don't get how my pug comment was confusing or illogical. Pugs are dogs. German Shepherds are also dogs. Pugs and German Shepherds look quite different. Thus illustrating the fact that two different animals can belong to the same group, despite looking very different.
Likewise, even though the animal in III didn't look like the targ we saw as Worf's pet, that doesn't mean that the III snimal wasn't a targ.
We've got this image, which we know for a fact is a targ:
We also have this CGI Targ from Enterprise:
Now, I think this creature is certianly similar enough to be in the same category:
After all, we have two creatures, both called targs, which are quite different. The first one is furry with just a few spikes. The second one has many more spikes, no fur and a large horny ridge on its nose. If two animals with such different looks can both be targs, I don't see why the Trek III critter can't as well.