I couldn't have explained it better myself. Glad you understand!

Aside from the last bit. The rainfall per unit area is not relevant to the shape of a particular container. 5mm of rain falls regardless of your container shape. Whether that 5mm of rain falling, also means that the water in your container is 5mm deep, is a function of your container shape.

It's not a unit problem. If you're trying to measure rainfall in a conical vessel, you can do that, and the conversion from collected volume to fallen rain will still yield the same 5mm out of the sky.