Is this supposed to have this amount of free movement

No. There should be a preload adjustment somewhere.

no, there isnt preload on that older style of crank, you either tighten the plastic locking nut on the non chain side crank and then tighten the pinch bolts, or if its already bottomed out you have to take the bottom bracket out and space the bottom bracket with the shims that comes with them.

If this is a shimano crank, watch a video for installing shimano hollow tech 2 cranks

fuck no