Why don't I have a custom title by now?!
Maybe a bit more, at least for what regards my country's football clubs ;-)
You're as qualified to answer this as I am.
Maybe of them do have a century-old history, and most started out as small regional or even neighborhood based football clubs, closely tied/knit with the local community, often a disadvantaged one or in hard times.
Some of them (AEK, PAOK) were founded by refugees of the Asia Minor Catastrophe, and so had a legacy of suffering, forced emigration and the bond that such situations cause. The football club was a sumbol for their lost homelands back in Asia Minor, and a lot of their energy and emotional investment now went to the club.
Other football clubs were more associated with syndicates/labour unions, or simply with the everyman's daily struggles. Belonging to the club meant belonging to a community that could offer support and protection (those were not easy times). Occasionally, the club had to be defended. An attack on the club was an attack on your community (see any similarity with Gangs of New York here?).
So, first and foremost, they offered a sense of belonging to mostly disadvantaged/poor/suffering/proletarian communities.
Now, AFAIK in the UK football firms were also often associated with a working class/blue collar/proletarian/disadvantaged background, for pretty much the same reason: they started out locally and as "grassroots" organizations in the aforementioned social environments, and membership had pretty much the same purpose and implications.
Now, why football and not, say, tennis or croquet? Unlike those other sports, football is a sport for the masses and the poor: even in the poorest 3rd world countries, you'll see children playing football with anything they can find, be it two stones as goalposts and a cabbage or rag ball as, well, a ball. So it's something readily identified, easily accessible, popular and easy to improvise. Something everybody can relate to.
The game of football itself is a team sport with numerous teams (most of any other mainstream sport), and has an explicit combat and "us vs them" element (though more controlled than, say, rugby). It's hard to feel the same way about a single tennis player, for instance.
Needless to say, a sense of belonging to a powerful group would be incomplete without "natural enemies" to fight and "victories" to be won. These elements are more easily channeled into violent behaviours (if anything, just for the herd/pack effects), and seeing how football clubs were proletar/blue collar to begin with, solving problems or perceived wrongs with your fists fitted right in.