Typically in the US, someone like me is a called a Democrat. Similarly, people who vote Republican are called "Republicans" rather than "Republican voters" or "Republican supporters."
In the UK, the dominant parties are Conservatives, Labour and the Liberal Democrats, while people are referred to as Conservative voters, Labour supporters, and so on. It's hard to even think of a noun associated with members of the Labour party (ok, well it's Labourites, but that's not used much). On the other hand, like in the US, party affiliation is also pretty stable, and there are still plenty of unquestioning supporters of each party.
But on the whole I like this use of language much better. Often the Democratic and Republican parties agree on things that the majority of Americans disagree with, and it'd be easier to talk about this if we didn't pretend that the entire country was made up of Democrats and Republicans.