Is it possible to have more than 1 profile at once? My son wants his own game, separate from mine.

HOMM doesn't use profiles, savegames are located in its own folder. Two installs will work if that's what you mean, but you'll have to use the copy-paste method and make a fresh shortcut.

Could you give a brief explanation of the copy paste method? Thanks

You could also create new directory (or two), where both you and your son will copy saved games. Just before playing copy your saves from that extra directory to game's saves and after playing copy those back where you have copied them from.

And by the copy-paste method he meant that you should copy the whole game directory somewhere else and then make a new desktop shortcut for that game's .exe file.