Not sure I understand. The city of London is in the county of Greater London. New York City is in the state of New York. But Singapore is a city-state.
In any case, there is no county of Greater London. Greater London is an 'administrative area' - have a read of the London Government Act 1963, which created it, and abolished the county of London which had existed before. That's why we have a Greater London Authority (and before that a Greater London Council), rather than the London County Council we had when London was a county. Of course, everyone treats it as a county because it would be maddening not to, but that's a bit like how everyone treats Washington DC as a state of the US (oh, we should add Washington DC to that list).
Legally, there is no 'London' which is a proper subset of Greater London (other than the City of London, which is a red herring in this case). So putting "London, London" is directly analogous to putting "Singapore, Singapore".
The New York case is different, but it still involves writing the name of your city twice. If anything, it's even sillier, because the whole state is named after one city.
EDIT: clarify about the GLA, GLC, and LCC (i hope)