I (american) say this every time I watch soccer/football which is usually every 2 years (olympics/world cup). I'd be interested to see the history/sociology of these chants and why they don't specifically extend to the US.
That being said, Hockey (NHL) seems to be a bit different. There are several team specific chants that have evolved but only usually after the home team scores. Every other non scoring chant is usually targeted at specific players.
Do people form organized fan groups for their teams? Because those tend to be the main sources of certain chants, especially against certain teams.
Side note: apparently there's no proper english word for a side-specific crowd? In portuguese, it's a "torcida" and the groups are "torcida organizada"
Does anyone know how all those songs got started? That would be interesting to learn about. Seems unlikely it just started with improv at the game, did a smaller group of people practice beforehand and then introduce them to the crowd?
In most European countries the Ultras are in charge of support. Some group comes up with a new song and the other people will start to sing it too eventually. Sometimes they even hand out small notes with the text for a new song. But also some songs go back decades and it's often forgotten how it started. Melodies are usually from famous songs e.g. Yellow Submarine, so it's pretty easy to memorize.
Since they don't have Ultras in England I can't really tell how it works there. I guess a bunch of people come up with something and if it's catchy it will spread quickly.
It seems to be largely a spontaneous rewording of existing catchy pub songs, from this one example. Could be related to Cockney culture which likes their rhymes.