I'm not aware about other countries, in Italy we have "minimum fuel" reported in our manuals and in our national AIP.
About the case you referred, you just had to give them delay as less as possible as of everyone else around but everyone who declared the minimum fuel, they can still fly normally, so it's like a PAN call (underlining the "like").
To avoid the problems there were in the past (cheating pilots
), someone should check them when on the ground.