HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

I would also like to note that you cannot save during your opponent's turn. It seems like a bug, as the message window indicating that the save was successful appears.

Without the mod, you can save during your opponent's turn, but it will not contain accurate data about your opponent's position. You cannot continue the game from this save, and you cannot analyze your opponent's moves after the game using it.
A complete save can only be made during your own turn. Therefore, I don't think it matters how the save function works during your opponent's turn in the mod.