At first I was going to contradict you but then I realised 10 mm of rain is actually a measure of volume or mass, corresponding to 10 kg, right?

And of course, people use it mainly as a rate, i.e. 10 kg per hour, or per four hours, or six hours, or 24 hours.

And it gets worse when we start talking about snow, the density of which can vary a lot!

Well depending on area you are counting it on. So yes per square meter it would be 10 kg. Your rain gauge might for example not have 10mm spaced markings...