I have a vertical monitor and want two full-width tiles/zones, but KDE is still trying to snap to a quarter of the screen. What am I doing wrong?

You have to hold the Shift key while moving a window for it to snap into the tiles you set up. If you just move them normally they have a different snapping behaviour like what you described.

Edit: So as the deleted reply was probably asking, this is how it works in full. If you have the KDE Plasma desktop environment after a certain version (I wanna say 5.6-ish?) you can do the following:

1. Press Windows+T, or as we Linux nerds like to call it Meta+T, to configure your "tiling zones" on your monitors.
2. Hold Shift+LeftClick on the title bar of a window to move them into the "tiling zones" you set up.

Discoverability on this sucks (as much of the Plasma desktop does) but it's a pretty cool feature.