EDIT: Sorry, you're referring to aquifers, not reservoirs.
It isn't like water in a stream because it stays where it is. It isn't like a reservoir or aquifer because the water isn't trivially dispatchable.
I suppose it depends where the snow is, some could be 'permanent', some seasonal melt, some gone by next week.
So really we should be differentiating the different types of snow?
SWE is monitored by satellites, aircraft, and by in situ measurements like from snow pillows. In California, the state DWR tracks SWE to estimate reservoir influx.
I trust that the experts know the difference between permanent snow and annual melting. They use satellite pictures to determine how much water will be available after it melts.
Edit: If someone knows more about the planning process than I do, please chime in.
You say "It all melts", and then "I trust that the experts know the difference between permanent snow....".
I hypothesised satellite photos, I'm not sure if depth of snow is relevant for more than just volume, and whether you could get good estimates of depth (and density???) from a satellite.