Inches or mm are a perfectly fine measurement for the amount of rain. Volume over area mathematically comes out as a unit of length.

It's a lot more intuitive with snow. If you get five inches of snow over night, there's no need for an area reference, because everything gets the same five inches of snow, no matter how big the area.

With rain it's the same, everything gets the same "height" of rain, even though it usually doesn't stay in place like the snow.