No, the car can't move side to side since the plates are bolted to the hubs. You could slide the car sideways if you pushed on it really hard but you would need to be on a smooth surface.

The plates I've built for myself are super solid. I can stand on the bumper and bounce the car to check the spring rate or to see how the shocks are working and stuff like that and the jackstands don't move around. You can't really do that with jackstands under the lower control arms since it isn't as secure.