http://pds26.egloos.com/pds/201408/10/94/f0006794_53e72b20438c8.jpg(Advertisement for korean version)
So this turns out to be a taiwanese game...A korean PC game magazine published in 2000 had an ad for this game, and I just thought it was a silly korean Pokemon knockoff.
The original taiwanese version is called 神氣兔寶貝 and was developed by Lonaisoft. Lonaisoft is still active and has a page for this game(
http://www.lonaisoft.com.tw/game.rabb.htm). Seems like Lonaisoft also made some Bubble Bobble knockoffs.
Korean version is called 내사랑몬스터(Means something like 'my dear monster', or 'my lovely monster') and published by Silia Entertainment. The Lonaisoft logo is still there so it seems to be an 'official' version for korean market. The ad above also mentions a game called 六道天書(another taiwanese game developed by different company) for Silia's next release.
http://pds25.egloos.com/pds/201408/10/94/f0006794_53e71f1469840.jpgThere's one more release for the game called 'Diki Game Collection 2', coming along with other three non-Lonaisoft games. This is published by another Korean company called Digital Dream Studio.
Both Silia Entertainment and Digital Dream Studio seems to be gone now. I can't track down Silia at all and DDS had its website
www.ddsdream.com and
www.dikigame.co.kr for its game division but both disappeared in 2003.