@A.Ellett in the comments points to the best explanation I've seen so far: this answer to the question, 「食肉目とネコ目、どっちが正しい言い方ですか？」
Ultimately, this use of ネコ in ネコ目【もく】 (literally, "cat order") as a label for order Carnivora, rather than the previous 食肉目【しょくにくもく】
label (literally, "eats-meat order") as a direct translation of Carnivora, appears to be a misguided Ministry of Education policy from 1988, which has met with (I believe understandable) opposition from the Japanese scientific community. Apparently some Ministry officials thought it made sense to choose a representative animal for each of the orders, and use that name in katakana for the order name, choosing katakana possibly out of a concern that the kanji were too difficult. At least some people in the scientific community have pointed out various concerns with this, including the following four issues mentioned in the linked Yahoo! Chiebukuro post:
- The clear meanings of the older names in kanji are lost.
- The concept of the order is narrowed (as in your very question: the name of the representative animal obscures and distracts from the many other animals that might be in that order).
- Since the higher-level species categories like "order" are broader and are already harder to understand, it is common to use the suffix 類【るい】 ("type, kind"). Using the katakana labels makes this more confusing, as these labels can be ambiguous: if we say ネズミ類【るい】, do we mean 齧歯目【げっしもく】 (order Rodentia, which includes lots of things that aren't mice or rats, such as the huge 50kg capybara), or ネズミ科【か】 (family Muridae, which is mostly mice and rats)?
- The new animal-name katakana labels have no connection to the Latin-based order labels used in the rest of the world. ネコ means "cat", and has nothing directly to do with "carnivora".
Perhaps my favorite bit from the Japanese poster's text:
Even if we use a specific animal's name, the biggest category where this isn't confusing is the genus, and it just isn't possible to represent a category as large as an order with an animal's name, and instead this causes confusion, but I guess the [Ministry of Education] people at the time in charge [of making this change] didn't know this.
Currently, it seems that both ネコ目【もく】 and 食肉目【しょくにくもく】 are regarded as "correct" in Japanese educational circles. In scientific circles, 食肉目 may be preferable, depending on your audience.