I just read (https://en.wikipedia.org/wiki/Michael_Johnson_(sprinter)) that he won his first 200 m and 400 m double at the 1995 World Championships in Göteborg, Sverige. I bet he heard that song there!
I took the trouble of listening to the song (not a fan) and you're right, there's no är (when I first googled it the lyric site I found actually did have the är). I think they really mean to say I, a song and dance man. This kind of construction works pretty much the same in both languages. You can leave out the är in headlines, when the phrase is an apposition (= appears in a position kind of like a title after the name, like in X, Duke of Y). In writing, you could have an – instead of the verb. There could also be a comma, eg: Du, en sång och dansman??? 'You, a song and dance man???'
So there are certainly cases when är/is will be left out. For Swedish one might even expect to see it left out more often in written texts than in natural spoken language where it would be so easy just to prolong the en. But of course these are song lyrics so there are many things in there that differ from natural language. More than the missing är, I think the extra en sounds bad: en sång och en dansman – the whole expression sounds silly to me, but especially the second en is jarring (to the pedant, it makes the sentence mean: I am 1. a song 2. a dance man). It's also very unnatural to say Det är knappt man sina ögon tror with tror at the end of the sentence.