Don't think that it is rust coming from iron water pipes (Must be old, copper or plastic these days)
The outer rim looks to so circular that it could almost be made by a stamp. Watch is good suggestion, but watch cases tend to be from Aluminium alloys or stainless steel, neither of which would produce what looks like a rust mark.
So something being rested on, or against, it while waiting to be washed?
Oxalic acid or a "stain devil" ought to shift it.