Speaking as an a tower and approach/radar controller, in my experience the request for a call of 'field in sight' (just to offer another variation in phraseology) is used for any or all of the reasons mentioned previously - but only in suitable circumstances. If there is a need to sequence the inbound into other traffic I would use different techniques. If, for example, the pliot appears to be having trouble navigating, I might ask for the call whereas in the same circumstances a pilot that appears to be on top of everything might get transferred earlier. Either way, I'd be keeping an eye on where the aircraft is in case the pilot doesn't do what I expected.
So, overall, for me there is no great significance to the call - I'll use it if it is useful in the pertaining circumstances, and something else if it is better.