One way to hold this: a party name is a label that can be worn by the organisation and by the people inside it, and English lets it slide between the two without changing shape. The grammar does the work that the words do not. 'The Liberal Democrats' with 'the' and a plural points at the party; 'a Liberal Democrat' with 'a' points at one person; and the bare singular in front of another noun — Liberal Democrat policy — behaves almost like an adjective. Capital letters mark the whole thing as a name rather than a description.