When one unit or support hability sais "Only once per turn" is the same that "only one time in my turn" or "Only once per turn, opponent's turn included"?

players alternate taking turns during this game, there is no such thing as a game round which would brake down into turns as it is the case in tabletop warhammer for instance

IOW a turn is one time througth the Kingdom, Quest, Capitol, and Battlefield phases, it does not matter whose the active player. A card that says once per turn also refers specifically to that cards effect, so triggering a second card with the same effect would not violate this rule. If the card leaves play in any fashion and is put back into play, even in the same turn, it counts as a "reset" and the effect can be used again.

So i can use the hability in my turn, and in my opponent's turn.

Yep.