Weakly, yes. Enough to hold its own mass at rest, but I don't think it would work with any type of acceleration. It will stick nicely to a neodymium magnet.
I'd expect sticking the tag to some ferromagnetic surface, such as a fridge door, by the magnet to have about the same effect as gluing the tag to that same surface would which I would guess would be none.
Generally in a permanent magnetic speaker the permanent magnet is solidly affixed to the speaker housing.
The part that actually moves to make sound (the diaphragm) has a coil attached to it. The signal is sent through that coil, which produces a variable magnetic field that interacts with the fixed magnetic field of the permanent magnet.
This interaction produces a force that moves the coil, and thus also moves the diaphragm producing sound.
You could probably design a speaker where the coil is fixed and the magnet is attached to the diaphragm, but that is generally not done. You want the moving parts to be as light as possible so that it doesn't take a lot of energy to rapidly change their motion.
Consider a speaker playing an N Hz sine wave--it has to change direction 2N times per second, and between each direction change you want it to move far enough to move enough air to for the sound to be easily audible.
If the moving part is too heavy you would need a lot more force to accelerate it enough to move far enough to move enough air in the short time you have in order to reach a specified loudness, and then it would take a lot of energy to quickly change direction and do the opposite movement. Hence, the heavy magnet is fixed and the light coil moves the diaphragm).
Microphones are similar. There are moving coil microphones, but I don't recall seeing moving magnet microphones.
You might think it would also be the same with phonograph cartridges, but there you do find both moving magment and moving coil designs [1].
The case I was thinking about was specific to the airtag. In this case a fixed voice coil moves a small permanent magnet attached to the diaphragm rather than the other way around (according to iFixit, quote below). So maybe a rare earth magnet oriented the right way could prevent the magnet from moving. There’s an air gap so it will have to wait until I can do an experiment.
Note from iFixit:
> Did you notice the “button” on the underside of the cover? That’s not a clickable button, like the Mate and SmartTag have, but rather the magnet we saw earlier in the X-ray. It sits right inside the donut-shaped logic board, nested into a coil of copper to form a speaker. You read that right—the AirTag’s body is essentially a speaker driver. Power is sent to the voice coil, which drives the magnet mounted to the diaphragm—in this case, the plastic cover where the battery lives—which makes the sounds that lead you to your lost luggage.