Why does Japan have an Emperor? Was Japan really ever an empire throughout its history? An empire is a sovereign state functioning as an aggregate of nations or people. As far a I know this has never been the case for Japan.

Japan is an Empire and has an Emperor because the British, French, Russians, and Austrians had Empires and Emperors as well.

Royal or Imperial “titles”, in modern historiography, are propaganda points to be utilised by sovereign nations in national myth-making. This is compounded by the fact that, per se, they're all rather vague in definition and inherently meaningless. Queen Victoria became an “Empress” because she didn't want her daughter to outrank her in Continental terms. No other reason. Similar to the Japanese as well.

Now the original title used by Japanese Emperors, back till the 8th Century, was simply “Great King of Yamato" and other Chinese-influenced terms, which came to be replaced by yet another Chinese term- “Son of Heaven”( also a generic claim for most Cultures over the World). As such, the Emperor is not as much a Head of State, as a spiritual Schelling point for devout Shintoists both as Head of the Yamato clan, as an expy for Head of all clans, as well as a descendant of the gods.

In other words, the “Emperor” of the Japanese is more akin to the Christian Pope rather than any political body per se; by the time we even get any real Japanese records beyond the odd-(probably forged)-mention, it's already the 8th Century and the “Emperor” is a puppet controlled by vested feudal interests. This is reflected in the difference in their terminology for “State/ Imperialism” and “Emperor”.

When the Boshin War ended and even prior to it, a key concern among several Japanese thinkers was the necessity to demarcate Japan and the Japanese people from European systems like protectionism, Christianity, and military dependency etc in order to prevent Japan from falling under the same predicament as other Asian states. The usage of the term “Emperor” in translations to foreign documents was part of such associated kulturkampf. As it is, the term is meaningless political dickmeasuring as we've already seen but sounds grand.

As for OP's doubt on the early history of Japan… technically, early Japan was an “Empire”, given that it ruled over many “peoples” with differing “culture”. However by this standard, ALL states were Empires; difficulty in transportation ensured widespread provincialism.

19th Century France was more of an “Empire” of different “nations” than Japan with French as we know it being a minority language, until Paris standardized/ culturally genocided the provincial cultures, depending on your stance of such matters. Japan's corresponding monoculturalisation- as far as such a term could be valid in a premodern society- hadn't begun over Honshu alone even as late as the 9th Century and was still ongoing in places like Hokkaido well into the 19th Century and even early 20th.

Does that qualify as “Empire”? Maybe. But by those standards, the Byzantines weren't an Empire from roughly after the 11th Century while any standard Indian or Chinese state was one from day one of inception.