after the end of the second world war, americans let the communists and socialists out of prison and threw prominent supporters of the prior regime (politicians and industrialists alike) into jail or banned them from politics, and so the left did fairly well for a while, winning a majority from 47-48. the right was splintered into several parties, among them the liberal party and the democratic party.

after 1950, the communist victory in the chinese civil war, the onset of the korean war, the red scare in the US and the general success of the japanese left at organizing strikes led to the "reverse course," where the american occupation essentially let the fascists out of prison and put some of the communists back in. the americans leaned on the japanese right to merge the two bigger parties on the right, the liberal party and the democratic party. after the 1955 election, that party more or less has run the country, either outright or in coialition government, with the exception of the lower house in 1993-1994, and in the upper house just now.

i don't know how they chose 'democratic' as a name for a bunch of old rightists, but 'liberal' was in the sort of european sense of the word, advocating a laissez faire economic policy, meant in contrast to the socialism of the labor parties.