There is a difference, and a substantial one at that, between a one-party system and a country whose politics have been dominated by a single party despite the existence of other political organizations vying for power.
China, with its total domination by the Communist Party and intolerance for competing political organizations, is a one-party state. The Soviet Union was a one-party state, with only its Communist Party allowed to be in power. Japan is a democracy, its constitution shaped by occupying American military authorities under the considerable leadership of General Douglas MacArthur. That Japanese politics have been dominated since the end of World War II by the Liberal Democratic Party does not mean that Japan is a one-party state. It is not. At any given time, there are between a half-dozen and nine or ten political parties free to contest elections in Japan.
Like all countries, Japan is complicated. Modern Japan represents a gradual merging of ancient and socially conservative cultures and a steady liberalization wrought by decades of existing as a democratic system. Its dominant political party is the Liberal Democratic Party (LDP), which is identified with conservative policies (though the LDP's positions on some important economic issues, like protectionism and insulation from outside economic forces, are more often characteristic of liberal or left-wing positions).
The LDP has been in power for almost the entirety of the post–World War II era because it has reasonably satisfactorily navigated the slowly evolving political and cultural transformations that have taken place across Japan during this period and because much of Japanese society remains culturally conservative. Evidence that Japan is not a one-party system, however, does exist. Twice, the LDP has been removed from power by the voters, especially during the early 1990s, when economic stagnation caused voters to reject continued LDP control and elect instead a coalition of opposition parties. That this coalition was unable to retain its hold on power was a result of its fractiousness, common to such coalitions, and to the fact that the LDP retained its overwhelming majority within the Japanese Diet, or parliament.
While Japan is not, technically, a one-party state, there is no question that its very prolonged time in power has given it important advantages over other political parties. With length in office, dominant political organizations are able to solidify their control over the bureaucracies that run government on a daily basis. Japan is, however, a democracy.