Similar to how you can do current sensing in motors

I do not think there are not enough variables here to make this work. Reading coil current won’t give you any information you don’t already know if you’re already controlling coil current. You need to be reading at least one more variable that is somehow related to coil position.

Turning off the coil drive and shorting the coil temporarily to measure current is unlikely to give you anything but the same current you were driving it with (minus some losses). Ie still not an extra variable.

you can send and receive with the same coil in a metal detector.

This introduces new variables! Reflected signal magnitudes (and distortions & phase delays) that depend on distance to a nearby metal object that you intentionally install near the coil (eg the metal casing).

Not sure how easy or reliable this would be to do in practice. I have my doubts but I could be wrong :)