Here is a song by Benny Carter, which I believe is originally from Duke Ellington: things aint what they used to be
What is the style of this song? Is it swing?
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"Swing" refers to a specific, characteristic rhythm.
This song uses that rhythm, although it is not heavily emphasized.
The most significant characterization of this song, however, is that it is a blues.
If you are a musician, I can explain the swing rhythm to you in specific terms using music notation. If you are not a musician, I suppose it's just something you have to learn to recognize when you hear it.
The original of this song is by Duke Ellington, who was one of the most famous artists in the "swing" genre (big band jazz dance music with a syncopated beat), and it definitely has the distinctive swing beat. This recording is from long after the swing era, but it stays relatively faithful to the original. I think it would be fair to call it a jazzy swing song.