No. You are refusing to acknowledge statistical differences between the sexes. Male toddlers get a surge of testosterone that females don't, so the sexes are differentiated at an early age.
Competitive games (wanting to win, keeping score)
Not sure, but I would tend to say the sexes differ with this, as girls prefer more communication and cooperation in their play
Physical exploration (climbing, jumping, testing limits)
Boys do more of this, with more risk-taking, that girls
Rough-and-tumble play (wrestling, chasing, mock fighting)
This is pretty much a boy thing
Object-focused play (building, vehicles, tools)
I would guess a slight difference, with boys preferring this a bit more than girls.
Just to make sure there is no misunderstanding, of course I am talking about statistics, so the inevitable outliers do not disprove the differences.