I cannot name a single Mexican team, and that is partly because the oldest club dates back to the 1940s. The oldest Brazilian and Argentina clubs date back to the 1900s.
Teams like Atlante and América were founded in 1916.
Pretty much, like in Europe, if you have any interest/talent in football, you enroll in your local club as a kid and go on from there, in Argentina you have a multitude of leagues until you reach Primera Division, so you have from 5 to 6 levels of competition in between organized directly by the Argentine Football Association, founded in 1891, also ran the first tournament in that same year, which makes it the oldest associated football league other than the British FA cup (1871).
Below that you have the regional/provincial leagues, the least populated province, Tierra del Fuego, has 2; Buenos Aires province has 70+ by itself.
I think the real mystery is, how come Brazil, Argentina and Uruguay play so much better than what you would expect from relatively poor countries?
My guess is that their leagues are fairly developed industries, like you would expect in the first world.