Question about Katarina

So, I've been playing league for a while, and I've been wondering something that's bothered me for a while...

Why is Katarina an AP champion?

Yes, I know she has some AD scaling, but it's pathetic compared to her AP scaling. No one builds AD on Kat. It's almost as bad as going full AP Garen.

Ability power is supposed to be like magic power, iirc, and nothing in Katarina's kit really seems "magical". Does she have magic daggers? Well, probably not. There's nothing in the lore about it, and everything but their shape seems to point to them being just... daggers. I guess bouncing blade is hard to explain, but it's like trying to explain Sivir's Ricochet. I guess the marks they place are a little magic-ey, but Zed's death mark is some mysterious form of magic, despite scaling with AD. I mean, the rest of his kit is AD, and Katarina's kit could be AD other than bouncing blades... I mean, apparently, you don't have to be magical to jump to a target (see Talon's Cutthroat), anyone can grab blades and spin them around their body, and anyone trained like Katarina could throw knives at multiple targets while spinning, if we want to talk about Death Lotus.

Katarina is also the only AP champion with costless abilities (aside from Rumble, but I don't count him because his heat system changes things up and does actually limit his spell casting). I'm also excluding energy champions (like akali, who usually builds hybrid) and Vlad (because his cost is health).

I know it may be a video game logic thing or a balance issue, but this really confuses me.