In a 2v2 if an opponent controls a battlefield with a Jayce and I charm their teammate's Akshan into the base that the Jayce controls, do they fight each other and both die?

No, this play is illegal.

According to RiftJudge FAQ #1089 and RiftJudge FAQ #260, it is illegal to move a unit to a battlefield occupied by a teammate. Teammates can never have units at the same battlefield. Therefore, you cannot play Charm to move an opponent's Akshan into a battlefield controlled by their teammate who controls Jayce.

Because the move is illegal, the Akshan will not move, no showdown will occur, and neither unit will die.