I wouldn't call myself fluent, as I don't really get to practice, but I can understand quite a bit of Japanese, and when I went and listened to a K-Pop song, I kept thinking that it was Japanese, but I couldn't understand one bit. Do Korean and Japanese share common roots? Is there any other background that would help me understand why they sound so similar?
Korean and Japanese are both language isolates, that is, languages not provably related to any other language. Okay, recently a dialect of Japanese has been moved to related language, but so far there are no ties between Korean and Japanese. A proposed Altaic superfamily would still leave them less connected than Gaelic and Sanskrit.
So your mind is messing with you.
What you may be hearing is a little sprachbund effect, where disparate languages approach each other because a lot of people use both. This usually shows up in vocabulary borrowing, so it shows how slight the sprachbund is in that you heard nothing reasonable.
You see, Korea was part of the Japanese Empire from the later 1890s, ending the Joseon Dynasty's rule, until the end of WWII. So they had half a century of Japanese influence, but of a hostile memory, so they may have purged any borrowings.