I think you are a
In chess and breath,
They use me to death.
Chess tournaments. No idea what a "breath tournament" is supposed to be.
Baseball, basketball, they like me too,
Though not as much as those, who do Jiu Jitsu.
Baseball and basketball are more likely to use league championships (MLB, NBA), while jiu jitsu tournaments are almost always knockout tournaments
From childrens games to mortal combat, I'm loved by all.
I can be seen in spring, summer, winter, or fall.
Games like conkers can have knockout tournaments, but so can blood sports. Tournments can be held all year round.
And let's not forget, those games of high bluffs,
Or of the rappers, like Hammer or Puff.
Poker tournaments, and rap battles
"4, 8, 16, 32 legs" refers to the number of competitors in a standard tournament bracket. An odd number of entrants is awkward for a bracket because some competitors have to have byes.
Anime shows are known for including tournament arcs, such as the Cell Games in Dragon Ball Z, or the summer tournament in My Hero Academia