Flying Ninja Cat has low range, slow movement and rapid, moderate damaging attacks with splash damage. She has one million health, being able to soak up many hits.
Strategy
In Ninja Awakens!, the support is light enough that you can win using only 3-4 meatshields (including Crazed Macho Cat and Crazed Wall Cat) and a bunch of long-ranged attackers (e.g., Dragon Cat, Crazed Bahamut Cat). Otherwise, use anti-Floating cats to deal with this enemy. Sadako Cat is great here, as she can tank the damage and weaken the Ninja. Cameraman Cat is another great unit because she outranges Flying Ninja Cat by 10. Just protect and spam her and the Ninja is killed in no time. Another method is stacking Necromancer Cats; if you use Sanzo Cat as well, you can halt Flying Ninja Cat almost entirely.
Her gender is confirmed by the Caltrops' description in The Burgle Cats: "Ninja Cat forgot to clean up after herself. Just rubber, but the pointy bits still hurt when stepped on! Imagine if it was actual steel!"
Her movement animation is identical to Kanzo Hattori's "Flying Squirrel Technique" in Ninja Hattori-kun, and her parachute looks just like Hattori's outside of having a swirl-pattern design. Furthermore, her Japanese name is Flying Squirrel Ninja Cat.
Incidentally, Hattori is afraid of frogs, which may be why Ninja Frog Cat only rides another Ninja Cat in a frog mask.