An airbrake applied on one side produces a drag along the longitudinal axis of the ship; it doesn't move the ship to the side but only applies a torquing force that produces yaw, which is a rotation around a vertical axis of the ship in the same vertical plane [geometrically speaking] as the airbrake but at the center of the ship. No application of the airbrake can produce a true sideshift in which the entire ship moves sideways equal amounts at both ends, in other words with no steering input, just shifting sideways. The game Killer Loop has a true sideshift like that.

Presumably, the double press of the airbrake button does not activate the airbrake at all, but controls another mechanism. The brake button is probably used just because the PSP doesn't have enough buttons to give each function its own button.

Lateral thrusters that apply side force in a balanced way could work on a Wipeout racecraft, and if the body of the thruster is pretty much fully within the body of the ship, with only the opening of the thrust jet exposed at the side, it's not going to catch on anything.