Today the Emperor of Japan is referred to in Chinese as the "Tian Huang", meaning roughly "Heavenly Emperor", which is just a Sinitic reading of the Sino-Japanese Tennou (the "ou" 皇 used here is not to be confused with the "ou" 王 meaning "king", which corresponds to Chinese "wang").
Although I can't claim to know, I would assume the Ming dynasty did not really acknowledge the existence of any foreign Huang 皇 or Di 帝. So did the Emperor of Japan style himself differently, perhaps as a Wang (like Korean rulers who IIRC used "Da Wang" 大王, "Great King"), or did the EoJ style himself a Huang while being referred to as a Wang in Chinese texts, or what was the case?
Related to this, were there any key differences in how the Japanese sovereign was styled compared to say, the Song, Yuan or Qing eras, in either Chinese or Japanese texts?
What's been quite interesting is how little I was initially able to find on this note – the Ming had quite obvious dealings with Japan in the 1590s, and the Qing from the 1870s onward, which you might think would produce something quick and easy in the secondary literature, but apparently that was not so. That doesn't mean I didn't find anything, and indeed I can go a bit further back. Obviously as this is sort of a list of titles, it will be a bit episodic.
The Annals of Later Han reference the provision of a seal to an embassy from the state of Wo in 57 CE, which was apparently recovered by accident in the eighteenth century by a Japanese farmer. Legendarily, the Japanese emperors were already reigning by this point, but the seal itself reads han wei na guo wang 漢委奴國王 – while its interpretation is not entirely certain, it seems to suggest a) that weina was a term for the main Japanese polity by this stage, and that the ruler was titled guowang ('king') as far as the Chinese were concerned. The use of a variation on wang as the title for the Japanese monarch repeated during the next major contact, when the ruler Himiko sent a delegation to the post-Han state of Wei in 239 that led to her being recognised as qin wei wowang 親魏倭王 ('Friendly to Wei, Ruler of Wo').
The next major milestone seems to have been in 607, when Empress Suiko's embassy to Sui Yangdi brought a letter which referred to both rulers as Tianzi 天子 (Child of Heaven), one in the land of the rising sun and one in the land of the setting sun. This led to a refusal to address the embassy, though contacts continued in an attempt to clear up the faux pas. Amidst the back-and-forth of 607-608 a new phrase had been coined, ri ben zhu ming le mei yu de 日本主明樂美御德, the last six characters of which are a transliteration of the variant imperial title of sumeramikoto (also written 天皇). On the Japanese side, this preserved the imperial title for their own emperor and elevated him above a mere guowang, while the Sui, and late the Tang, could still interpret the title as placing the Japanese ruler in the inferior position. There also seems to have been a variant form of that beginning with xu instead of zhu but I haven't yet found the character for it.
What complicates things, though, is the use, real or proposed, of ri ben guo wang 日本國王 ('guowang of Japan') to designate the de facto head of government at various times. Taira no Kiyomori, daijō-daijin under Emperor Takakura, was referred to as such by an embassy from the Southern Song in 1172; during the peace negotiations of the lull in the Great East Asian War (or Imjin War), the Ming's final terms to Toyotomi Hideyoshi in 1595 also included his recognition as ri ben guo wang – prompting him to angrily assert that he was the king of Japan for all intents and purposes, whether the Ming recognised it or not! And of course in between, the Ashikaga shōguns had also been referred to by that title. As the principal recipient of any correspondence would be the functional ruler rather than the ceremonial, it seems as though the title ri ben guo wang remained standard, and applied to emperors or shōguns depending on who was actually in charge.
This changed in 1635, when Tokugawa Iemitsu first began to accept the title of [ri ben guo] da jun/[nihon-koku] taikun [日本國]大君. This came after the alternatives, both wang and jiangjun/shōgun, had been rejected by Iemitsu's predecessors, on the basis that wang both implied subordination to China and eroded the status of the Japanese emperor, while shōgun did not really translate well (as its Chinese reading, jiangjun, simply means 'general'). Taikun ('Great Prince', from which English loaned the word 'tycoon') got around it by being a distinct title of majesty that did not challenge the status of the emperor within Japan, while also not having precedents that implied subordination to China.
I've had less luck finding any specific information as to how the Qing viewed the 'restored' Emperor Meiji after 1868, but it seems from the 1871 and 1885 Qing-Japanese treaties that the Japanese imperial claim was at least tacitly recognised, as the treaties include both the Qing and Japanese reign dates. While some in China asserted that Meiji's title was illegitimate, particularly in the wake of military defeats to Japan in late 1894, the ultimate Treaty of Shimonoseki gave the two rulers equal titles and honorifics: da qing di guo da huang bi xia 大清帝國大皇陛下 and da ri ben di guo da huang bi xia 大日本帝國大皇帝陛下 ('Great Qing/Japanese Empire | Great Emperor | His Majesty'), settling the question once and for all. Well, at least until the Qing were overthrown and the Republic of China installed in their place.