After raising a car for inspection, I noticed that a mechanic gave all 4 wheels a spin.
What exactly are they looking for?
I assume it has to do with whether the wheel on the opposite side will spin as well? Or is it more about listening for any strange sounds? Is it related to wheel flanges or bearings?
Probably a combination of things - as you surmise, it's a good way to check for worn bearings (you may also spot them rocking the wheel up and down or side to side, that's also checking the wheel bearings).
Spinning them can also show up a binding brake (as the wheel will be hard to spin), or a problem with the drive train, such as a knocking CV joint on a front wheel drive.
You should also see them lift each wheel with a tool that looks a bit like a pry bar, that will be checking the suspension bushes for excess play.
What exactly are they looking for?
Anything that could be wrong or irregular.
Mind you, if this shop also does repairs, this is also a sales function: they are developing a list of repair items to recommend to you, almost everything on the above list is a "hundreds of dollars" sort of repair: high enough to make real money but low enough most people will say "sure, yeah, do it" when asked. That list is a mainstay for shops - even electric cars have all those items.
Lastly, if part of it is putting the car on a chassis dynamometer (as for some smog inspections), they are looking for any surprises - e.g. something that would make the car lurch off the dyno and careen across the shop, or fly at their head once the wheels are spinning.
I assume it has to do with whether the wheel on the opposite side will spin as well?
No, they don't care about that. Non-driven wheels are totally independent. Driven wheels have a differential between them which allows free motion between drive wheels (so you can go around corners without tires squealing)... so that won't tell you much either. "Spin opposite", "spin with" and "don't move" are all normal results when spinning 1 drive wheel with vehicle in neutral.
