System Information
Operating system: Windows-10-10.0.19044-SP0 64 Bits
Graphics card: NVIDIA GeForce GTX 1070 Ti/PCIe/SSE2 NVIDIA Corporation 4.5.0 NVIDIA 512.15

Blender Version
Broken: version: 3.1.2, branch: master, commit date: 2022-03-31 17:40, hash: rBcc66d1020c3b
Worked: untested

Short description of error
After assigning object A to a modifier field on object B, object A cannot be snapped to any faces/edges/vertices/etc of object B.
Example: When using an empty as a projector for the UV Project modifier, this empty cannot be snapped to the faces of this object.

Exact steps for others to reproduce the error

1. Create an object
2. Add a modifier to it which has an object field
3. Create a second object and assign it to the modifier

-> The second object now cannot be snapped to the first object