I think Shimano uses 8 strain gauges in the pedal axis and after you have installed the pedal it auto-calibrates itsel after a couple of revolutions based on the knowledge of where in the rotation cycle the rider will be able to push the highest force.

Bike computer systems normally also make use of a cadence sensor, to measure number of revolutions/minute of the cranks. Such a sensor would also work as an index marker when the cranks are at a specific rotation angle. So it would be quick to notice which of two measurement channels that represents the left and the right pedal.

Some cheaper systems only measure force on one pedal or a combined force for both legs. Some systems (what I would consider "better") separates the readings so they can tell what imbalance there is between left and right leg, besides the absolute power value.

When it comes to forces not driving the cranks, I don't think too many cyclists have the energy (or stupidity) to waste any larger amount of effort into trying to push the pedal in directions that the pedal does not move. Standing still waiting at a red light, you may push down on the bottom-most pedal but with zero RPM that would represent zero power. And if same thing is done while biking, a system that can guestimate an approximate angle of the cranks and an approximate angle of the force (multple strain gauges) would know there is a zero momentum axis at that position. So it would still not compute it into power.

In the end, you must have multiple strain gauges unless you can guarantee that the strain gauges are always arranged so they see maximum deflection from a tangental force and not a radial force. If you can manage that, then the physical sensor will mostly ignore forces in wrong direction. If you can't guarantee that and needs multiple strain gauges, then you automatically gets a way of computing the direction of the force - so it can be splitted into a radial and a tangential component.