I'd posit that neither is particularly healthy.
Keep in mind that "standing" in this case means standing in one place on a hard floor. If you're comparing to the typical tech worker standing-desk setup, most such workers have done a lot to get the ergonomics of that setup correct, i.e. getting surfaces and screens at the right heights and adding a pad to stand on, whereas the ergonomics of a cashier station are geared toward the customer, not the cashier. You could address this partly by adding pads to standing surfaces.
There's a larger problem here though, which is that the evidence behind standing desks being beneficial is... not good. There's lots of evidence that sitting is bad, but only in comparison to not sitting. That hides the fact that "not sitting" includes a lot of different kinds of standing--walking around is not the same as standing in one place. What studies we have of standing desks document back and leg pain as a possible side effect, and that's usually the best case of standing in one place. If there are any studies of standing in one place on a hard floor for hours, I'm not aware of them, but I'm not convinced that particular case of "not sitting" is much better than sitting.
In short, the problem with sitting is probably that you're not moving, not the particular position in which you don't move.
Talk to your cashier, ask how long they've been working there and if they experience pain from standing, and with the exception of new employees or employees under, maybe 25ish, I can almost guarantee the answer will be "yes".