This enemy has mediocre health and insane DPS, so you have to treat it like an R. Ost, with a lot of meatshields and rangers. This enemies biggest weakness is low range and low speed.
Due to Manic Island Cat's high DPS but somewhat low-ish health, the best way to deal with it is to spam area attackers that can outrange it like Cyborg Cat or Drama Cats. It will still probably advance and kill some of your stack, but killing a stack of Manic Island Cats provides enough money to keep your spam of area attackers up.
Ogre Island is the only stage in which Manic Island Cat appears in, which has a Berserkory in it. Naturally, any Wave Shield units will be good to bring, some examples being Hatsune Miku: MM2020 Tokyo or Eva Unit-00. Note that Wave Shields with lower range or close range to Manic Island Cat's will be shredded extremely quickly due to their high DPS.
This enemy has always appeared at 100% strength magnification.