Player character handles objects and tools equally well with either hand, automatically switching hands as situation requires or offers a method to quickly switch between them.
The first video game about Ambidextrous was released on March 29, 2005.
Ubisoft has published all these games
* Dual wielding - allows wielding two items at a time in each hand, but may not allow using the secondary hand alone.
* Cover system - trying to use items from the secondary hand side tends to reveal your character much more than from the primary hand side.
Should NOT be used on games where this has no impact on gameplay. Such as an RPG which allows placing the tool in either hand in inventory but in use it's either always on the right, or the left hand use has no practical differences.
Likely involves the camera swapping to the other side of the character in 3rd person games.