You should not, and I can't feel any play with the 3 bikes I've ridden with through axles.

However, in all of them experienced what the OP is saying. Play may be imperceptible, but it's there and, if the rotor is close enough to the pads, it can make it rub the pads.

In my opinion, the issue is that the hole where the axle goes through is obviously a little bigger than the axle itself (otherwise it wouldn't fit) and allows some degree of movement. It's not that the axle is moving constantly, it's more like it slips to a slightly different position when you start riding the bike and stays there. And in my experience, it happens even with a really tight axle.

I have experienced this with a Suntour fork with Q-LOC axle, with a Fox Rhythm 34 fork, and with the rigid carbon fork on my road bike.