This is not true. Try listening to "the same" song in two different languages that you can speak, one of which being the language the song was written in.
One of two things will be true:
(a) It is obvious which songs was originally written in that language and which is a translation.
(b) It is not obvious that the two songs are the same.
> I believe the correct term would be lyrics, not songs.
Nope. The lyrics are the song. Music with nothing but lyrics sung to a melody is still a song. Music without lyrics is not a song.