The "myth of fingerprints" is that by focusing on trivial aspects of ourselves, we appear to be very different when in fact we have much more in common with one another. It's also a literal myth: fingerprints are highly differential but they are not unique.
As user @PeterShor very helpfully observed in the comments, our commonality is emphasised by this line, where "they" is fingerprints:
I have seen them all, and man, they're all the same
While our differences are stressed by the final verse:
Lives a former talk-show host
And far and wide his name was known
He said, “There’s no doubt about it
It was the myth of fingerprints
That’s why we must learn to live alone”
The myth of fingerprints condemns us to live alone - as atomised, quarrelsome individuals - when we could instead enjoy the fruits of living in harmony. It's interesting that the subject of the song is a talk show host, someone who makes a living interviewing the rich and famous and thus focussing on what makes them "different" from everyday people.
The previous verse becomes interesting in this context:
Out in the Indian Ocean somewhere
There’s a former army post
Abandoned now just like the war
And there’s no doubt about it
It was the myth of fingerprints
That’s what that old army post was for
Armies are required because we fight over our differences: it's what "the old army post was for". There is, however, hope that we might learn. The army post is abandoned. There is also only one US army base in the Indian Ocean - Diego Garcia. Its inhabitants were forcibly removed by the US navy prior to building the base.
It's worth remembering that this is the closing track on an album which features critiques of racism as a major theme. Racism is founded on differences - the myth of fingerprints. This is all over other repeated structures in the song. "Watermelon" is a common racist motif in the US. "Black pit town," could be a mining town, or also a mining town populated by people of colour.
The lyrics about the sun rising and falling ("bloody" - another conflict motif) and "all around the world" suggest this has been going on essentially forever. We notice it - "somebody says what's a better thing to do" - but do nothing.
So this is a song about how conflict and oppression are built on emphasizing small differences between people. It opines that this is human nature and that we must, to some extent, learn to live with it. But that it shouldn't stop us asking questions and doing what we can to improve matters, even if we must, eventually, "learn to live alone".