How does an 'equal division in spermatogenesis' result in sperms competing with each other? "The division happens asynchronically; if the tube is cut transversally one could observe different maturation states. A group of cells with different maturation states that are being generated at the same time is called a spermatogenic wave" (from Wikipedia). This doesn't sound like any chaotic or even violent competition to me? And there is a asynchronous division in spermatogenesis.

In contrast: "primary oocytes reach their maximum development at ~20 weeks of gestational age, when approximately seven million primary oocytes have been created; however, at birth, this number has already been reduced to approximately 1-2 million per ovary. At puberty, the number of oocytes decreases even more to reach about 60,000 to 80,000 per ovary, and only about 500 mature oocytes will be produced during a woman's life, the others will undergo atresia (degeneration)." (Also from Wikipedia)

So you could actually reverse this meme! Spermatocytes have a way to cooperate who goes first because only some are already fertile. On the other hand, oocytes really have to fight for their chance to ever get a go or else be degenerated.

So why is this meme not reversed? Because people tend to project their own norms and expectations onto everything! And this leads to a lot of wrong assumptions. People arguing about sex and gender often use these concepts of biology and what is 'natural'. But these are all only really projections of their expectations. Animals are not abiding to our gender norms. There aren't only male or female animals. Even archaeology has been heavily tainted by sexist assumptions on who has to be a man or woman.

This meme, just like bad science, is reproducing sexist assumptions. And now people who see this meme will think that there lies some truth within it, because the joke wouldn't work otherwise. So they'll assume that sperms really do compete for their opportunity while oocytes negotiate. But this isn't a fair or correct simplification of what's actually happening.