You must remember thisThere's assonance (with "i") in the second line - "A kiss is still a kiss." There's an insistence and prevalence of that "i" sound, which emphasizes the constancy inherent in the meaning of "still."
A kiss is still a kiss
A sigh is just a sigh
The fundamental things apply as time goes by
Furthermore, that same "i" sound is in the "this" from the previous line. The whole line "A kiss is still a kiss" is linked to "this" grammatically ("A kiss is still a kiss" is part of what "You must remember"), but they're also linked through that assonance. There's the "i" in "this," and it's widespread in the line "A kiss is still a kiss."
The third line is interesting in its lack of that insistent assonance. In "A sigh is just a sigh," that "i" sound is present only in the verb ("is"). In the two nouns ("sigh"), it's changed from a short "i" to a long "i," and there's a complete lack of any "i" (short or long) in the adverb ("just"). Because this line doesn't have the same repeated vowel sounds as the previous line (or any consistently repeated vowel sounds at all), it's slightly disappointing, so - compared to the kiss that "is still a kiss" - this line has the same unimpressive quality as a sigh, which is "just a sigh."
I should note that in the movie Casablanca, it's sung as "A kiss is just a kiss," which decreases the strength of the assonance-linking present in "A kiss is still a kiss." I lookt up the lyrics for some other versions, and it seems like some have "A kiss is still a kiss" and some have "A kiss is just a kiss."