It's an intentional attempt to put the Japanese monarch on the same level as the Chinese one. It's even true in the Japanese language; (天皇 tenno/tianhuang )was chosen as a gender-neutral term combining 天子(tenshi/tianzi, Son of Heaven) with 皇帝 (koutei/huangdi, the title usually translated as Emperor).

Basically it's an assertion of equality between the Chinese and Japanese monarchies. The Chinese, as you can imagine, were not too fond of the term.

Others have covered the Japanese, now let me give you some perspective into the nomenclature of Thai kings as a Thai.

Thai monarchy is and was highly respected historically, to the levels of veneration as deities. It is culturally significant that there a multiple levels within the Thai language, and the words that you would use around the Royal family are entirely different from those those that you would use colloquially, in order to show proper respect.

In the time of Rama I, who established the current line of Thai kings, you would not refer to him by his name. His name was seen as sacred, and therefore different methods of referring to his majesty were used. Buddha Yodfa Chulaloke, as the English name references, you could see that Thais were already equating his majesty as a Buddha, or Enlightened One, in reference to our National Religion. He was also called the Thai equivalent of "First Ruler" at the time, แผ่นดินต้น. His son was called "Middle ruler", and the third... well by this time the Thais realized that they didn't want to call Rama III แผ่นดินปลาย, or the "Last reign". Thus there was an interesting period in which the third, fourth, and fifth King of Thailand had titles that were self-administered or given posthumously.

The fifth king of Thailand is the famous Chulalongkorn, who is beloved for keeping Thailand free and strengthening the Thai state while the surrounding countries became owned by various European powers. Even though it's not historically accurate, it's a useful reference for farang (foreigners) when I state that his highness Chulalongkorn is the prince in the musical "The King and I" along with his father King Mongkut. Both were scholars who studied Western society in order to protect our country and people from the encroaching Western powers destroying the surrounding countries' cultures, livelihoods, and freedoms, and Chulalongkorn abolished slavery and led reforms in the modernization of Siam. King Mongkut and King Chulalongkorn the Great set the groundwork for adopting British culture and ultimately British-style government. Therefore in translation to English, the Thais used the British name for their supreme monarch, King.

Ultimately it wasn't until the reign of King Vajiravudh, or Rama VI, that the nomenclature of Rama (roman numeral) name was taken and popularized, in an effort to easily refer to the Thai Kings similarly to their Western counterparts. The Rama name itself is taken from the Hindu god, Rama, as Hinduism has a large influence on Buddhism and Thailand. This practice continues to this day with King Rama IX Bhumibol Adulyadej the Great, currently longest reigning Monarch in the world. Long live the King.

I hope that answers your question from the Thai side. Anything else and I will try my best to answer. Edited for clarity.

"King" and "emperor" are both European concepts, while the Japanese and Thai of course have their own words for their respective monarchs.

Sorry for the comment, but I don't understand this at all. Neither of those two words represent in any way "European concepts." Non-English speaking European people also have various other words for "monarchs".

Can you clarify your question?