It's important to remember that planes are essentially infinitely thin. So, when two planes intersect, you end up with a one-dimensional line without any visible faces. Even if you try to use a 2D plane with a solid object, it may not work as intended because the plugin is designed to work with solids.
Consider the types of output you'll get when different geometries intersect:
If you need a plane as an output, it's recommended to intersect with a very thin box, like a 2 x 2 x 0.01, instead of using actual 2D planes. Think of it like using a cookie-cutter.
Furthermore, ensure you're working with manifold geometry rather than non-manifold. You can find more information on this topic here: