I never worked with these before, but these coils are apparently called "Voice Coil Motors" (VCM). The one from this post is big but its the same tech that is used on a smaller scale in phones.

I went and found a "driver" for these things and took a look at the datasheet to understand how a VCM is typically controlled. This thing is 1.5mm x 0.8mm small so something you might actually find in a phone.

The driver chip has built in current sensing and the current going through the coil directly corresponds to lens position so just from that, the driver knows where the lens is positioned.

https://www.ti.com/lit/ds/symlink/drv201.pdf#page=8

The device is used to control the current in the voice coil motor (VCM). The current in the VCM generates a magnetic field which forces the lens stack connected to a spring to move. The VCM current and thus the lens position can be controlled via the I2C interface and an auto focus function can be implemented

Because the lens stack is spring mounted it can lead to "ringing" which basically means bouncing that continues after the desired position was set. So the chip apparently has built in logic for compensating for that so it more smoothly reaches the desired position without overshooting.