Whenever you talk about "always" in terms of human endeavors, the answer is likely to be a resounding "NO!". And so with this question: No, songs don't "always" have to make sense.

Indeed, not infrequently the point of a song is in its nonsensicality (if there is such a word).

In my own case, I predominantly sing story songs, and of course in order for a song to tell a story it has to make sense, at least to a degree. For a singer who likes "Ooop, Ooop, shoobee dabbah" sort of thing, it's a different matter altogether.

"Whatever floats yer boat," as some great genius once said.

Dave Oesterreich