My son is playing Fall Baseball right now, but his last game is at the end of October. Two years ago he was playing for the same league in January or February...was that Fall Baseball as well, or is there a Winter League?

yes, there is winter baseball at some places

There are both fall and winter leagues. My highschool has a fallball program, and the little league is having winter leagues.

If your kid really wants to play in both, and you find another winter league, I'd sign him up for it. There are lots of guys that make it to the MLB (most out of the major 3, NBA/NFL/MLB) so he can have a good shot if he really wants it.