Whilst there are plenty of very genuine scientific inaccuracies to argue about in the bible, I honestly don't really think this is one of the better lines of argument. I'll explain why:
Taxonomy isn't absolute, since the groupings themselves do not exist in nature, but only within the taxonomical system itself. In other words, Taxonomy is a way of sorting and grouping similarities and differences between biological specimens (and nowadays, particularly as is informed by evolutionary biology). Or, to simplify further, taxonomy is a system of measurement. One that, notably, has changed and evolved over time itself (much like the biology it studies!). Originally, the system was not meant to allow for certain groupings to exist within the exact same type of group... in other words, you weren't meant to have a "Genus" inside of another "Genus", or a "Class" inside of a "Class". However, evolution isn't simple, nor neat, so nowadays, we have situations like "Aves/Birds", which is a taxonomical class, being inside of "Reptilia/Reptiles", which is yet another class. And furthermore, terms we've inherited from the ancients like "Fish" don't even really mean anything taxonomically, despite being a commonly used word applied to animals, which is a quirk of how cladistic taxonomy works.
Now, let us assume that God does exist... probably not the God that folk Christians traditionally imagine, but we'll go with it. Who is to say that God would really make these taxonomical distinctions? They were invented by humans as a form of very useful, but ultimately inabsolute measurement system for biology.
I am reminded of the quote by Sartre "Existence proceeds Essence", or as Deleuze similarly puts it: "Difference proceeds Identity". It is to say that the labels we use for things are merely helpful categories created and retroactively applied after the thing exists already. Of course, this is a philosophical question... one might believe that the categories are actually innate to reality, but personally I find that unlikely... So who is to say God doesn't view biology with this lenses? If that were true, and God was directly writing the Bible, then it may make a lot more sense to use the arbitrary language of the time, rather than the arbitrary language of the 21st century / future.
Plus, carrying on, as user @enfys points out... plenty of people don't believe the Bible was actually written by God, which is actually textual. The Bible does not claim for itself that it was... It's a pretty common folk Christian tendency, and especially Protestant tendency, to act like this is somehow a textual idea whenever it isn't.
So yeah... I think this line of argument falls flat in multiple regards.