I'm not sure of the details and precisely why the taxi industry has so much power. (Maybe they have carefully compiled dossiers on the late-night trips of every politician in the city...)
But medallion owners care. If someone says they will end the program, they will vote like crazy against it. They will also tell the drivers (who might actually stand to gain if they could get their own medallions at a lower cost) that margins will drop, cheap competitors will move in, safety standards will drop, and so on.
The same is true of sugar imports, patent laws, retrospective copyright extensions, and other issues which have a few people who care about it, and a lot who don't even understand it.
OK, you could say that deregulation is what the right (or libertarians) stands for. But sometimes it works the other way - regulations and subsidies can arguably help the broader community while hurting a few special interest groups.
Hillary Clinton used to be a senator for the state of New York and now is the Secretary of State of the US. Neither of these roles have much to do with New York City government.