Source? Every source I've seen says that MM is derived from the Roman numeral M, meaning thousand.

Wouldn't that make MM 2,000, the same way II is 2 and XX is 20?

It would be if it was interpreted as an actual Roman number. But in this case it's treated as M x M.

The actual Roman version of a million is an M with a bar over it, where the bar means x1000. But that's not an ordinary character, so wouldn't work for this purpose.