As far as I know, prior to the adoption of the title "Emperor" in ancient China by Shihuangdi (who claimed to be the first 皇帝 (huangdi), from the titles of 8 ancient godly beings), 王 was the sole title for sovereigns. After that, 王 was still used for certain nobles to some extent (such as Prince Wucheng in China or Prince Hưng Đạo in Vietnam), who were somewhat akin to European "sovereign princes" (known as Fürst in German), such as those of Monaco or Liechtenstein. 王 was also used by monarchs of Korea, most of whom didn't claim the title 皇帝. However, I'm not aware of many instances of 王 being used for "son of the Emperor".
In Japan, an imperial prince is titled 親王. It sort of makes sense that he's not titled 王子 ("king's child") because we're talking about the Empire, not the Kingdom, of Japan, but then 皇子 ("emperor's child") should also be a valid alternative given the Emperor of Japan is still titled 天皇 ("heavenly emperor").
The Japanese Wikipedia article on 親王 claims (with no sources) that the title 親王 was adopted in all the Sinophere countries (China, Japan, Korea, Vietnam), but only lists Japanese and Manchurian 親王's. I'm not particularly familiar with Korean so I haven't been able to check if there were 親王 in the Korean Empire as the article claims, but I'm pretty sure there were only Manchurian 親王's in China (other 王's were more "kings" than "princes", and even in cases where they were "princes", they were still titled "sovereign princes" by a Chinese emperor, they couldn't just suddenly become his "sons"), and there were no 親王's in Vietnam. Most of the time, sons of emperors went by 皇子, or 太子 if they were crown princes (first in the line of succession to the imperial throne). In fact 皇子 and 公主, the titles of imperial princes and princesses, have long come to mean any kinds of princes and princesses (emperors'/kings'/sultans' sons/daughters, the husband of a Queen of Great Britain, excluding the wife of a Prince of Great Britain but including a (sovereign) Prince of Wales) in Vietnamese.
Main question: So what's the origin of the title 親王? Given the long history of the Empire of Japan, would the possibility that the Japanese adopted this title from the Manchus likely at all? And what is its etymology? Does it mean "familial 王" or "parental 王"? And how come the word 王, which was used to refer to "sovereign rulers", came to to refer to "sons of emperors"? This Chinese Wikipedia article claims that the title 親王 goes all the way back to the Spring and Autumn period in China, is that true?
A related, and even stranger title would be 女王, which definitely comes up in your typical bilingual dictionary as meaning "queen (regnant)", but it is in fact also the title of a Princess of Japan (a prince's daughter, as opposed to his sister who's titled 内親王 and his consort who's titled 親王妃).