(This works with both the unpatched and 1.0X.2 versions.)

Is this a bug or a game feature which I don't understand correctly?

You can move any ship two times minus one movement point each turn when (ab)using the autopilot.

E.g. the flagship has three movement points in the beginning of the game. Spend those, select the ship again and set it to move two squares further. Next turn it moves those two but when you select it again it still somehow has all three movement points left.

I first noticed this many days ago but thought that I just didn't understand how the game works. Now, I just don't get it. Am I missing something?

What you describe is not possible because the autopilot always moves at the end of the turn (exception: The turn where you give the order).

So whenever you select an autopiloted ship, it hasn't moved yet and therefore can still be moved this turn.

So..... either your description isn't accurate (are you sure you didn't have one turn more than you think?) or something else is quite wrong.