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Liters / m^2 is common unit for the amount of rain. Other unit is mm/m^2 for rain. The units are exchangeable

Yep, it's pretty normal, though I also often see just plain mm and many weather apps/sites often only display the probability of rain.

Yes, as others have said, this is the standard outside of the US. The neat thing about metric units is the ability to convert stuff more easily.

If we take our rainfall example here: 1 sqm is a square with 1000 millimeters (mm) on each side. To go one order of magnitude up you just divide by 10, so you get 100 centimeters (cm). This is not necessary, but it is more common to calculate small liter amounts in cubiccentimeters (cm3), that's what the cc/ccm on measuring jugs or the volume measurement of engines stands for. This is usually referred to as milliliters (ml). 1000 of these make up 1 liter (l) (that's where the biggest irregularities of the metric system lie

So image that 1 sqm square in front of you (the scale compared to feet/inches isn't that important). To fill that square up we would need 10.000 of our little cubiccentimeter-cubes, 100 on each side, 100*100=10.000. If we now fill up our little cubes with water we raise the water level in our square by 1 cm/10 mm. 10.000 ccm/ml are equal to 10. So 10 mm of rainfall would result in 10 l of water, 25,4 mm (1 inch) would be 25,4 l and so on. As I'm not familiar with American weather reports I don't know how granular the reporting is if you have to work with fractions of an inch, but I guess down to 1/10 would still work for me, but don't tell me to imagine something like 11/17" or somesuch.

That's what makes the metric system so great. If you understand the underlying mechanic you can easily convert everything.

This would still work in imperial units in principle, but because length and volumetric measurements don't align properly you would have trouble converting them in your head. 1 gallon would be 231 sqin, 1 cubic foot would be 7,481 gallons. There is no way to fit these into a system where you could easily convert the height of your water level into a volumetric measurement and that's why the US sticks to just using inches for rainfall.

It's a global standard. You're in Murica.

Maybe the us national weather service just assumes a volume and doesn't say it? They kinda have to. If you say it rained x inches of rain in that hour, how do you know in what area? X inches over the whole country? Or per city? Per square inch? They have to measure the rainfall over a certain volume over a certain time. Otherwise you wouldn't be able to compare it.