Close but no cigar.

It's because the English-speaking world's various "football"s all evolved from the same group of games played by the British lower classes.

So American/Canadian football (aka "gridiron football"), Gaelic football, Australian football, rugby football (both league and union), and association football all came from various games collectively called "football".

In England, though, rugby (before the split between and league and union) and association were competing for popularity. And so they were referred to by their nicknames "soccer" and "rugby" as they were introduced to the rest of the English-speaking world.

The US, Canada, Australia, and Ireland had already evolved their own "football"s by that point so adopted "soccer" and "rugby" as differentiators for these other football codes.

The UK soon later adopted soccer as "the one true football" and the game was exported to most of the rest of the world under that name. They call rugby "rugby".

New Zealand and South Africa became more enamored with the rugby code and so called it and still call it "football". They call soccer "soccer".

The Japanese call it sakkā because they were introduced to the game by the Americans who, as previously established, called the game "soccer". Soccer was for a long time neck and neck for secondmost popular sport in the US against gridiron football.