From the Latin ambo, which means 'both', and dexter, which means 'right'. As the right hand is typically the one to possess superior control, it is the one associated with the quality of dexterity, hence the name. Ambidexterity, then, is not having two right hands, but having each hand capably dextrous.

Speaking from a personal viewpoint, being ambidextrous can come in handy, (no pun intended). Nevertheless, absolute ambidexterity is almost unheard of. Usually, it is only some number of qualities that achieve true either-handedness. Myself, I can manage many things with either hand. Yet there are some things that are comfortable only with one hand or the other.
For example: Bowling ball, right. Hockey stick, left. Rifle, left. Sidearm, right. Just whatever feels correct.

I have to admit, though, on some level, I do enjoy watching the facial expressions when I switch hands in the middle of writing something. Does that make me a strange person?