Between straight men, masculinity is definitely asserted through desirability to women. You're more of a man compared to your male peers if you can "get" more women.
no one is saying men don't compete with each other. But everyone knows they are better at being friends than women.
Never have I been anxious around bringing 2 of my male friends together. When I introduce a female to another female, I'm always walking on egg shells.
I don't see it.
It reads like another social stereotype
Strength, dick size, athleticism, as Sunshine stated.
Also metrosexualism is becoming more of a thing so you could also argue for fashion, grooming, etc.
Do you think men actually talk about their penis sizes amongst each other or compare? I would argue it mostly comes from women and their preference for large penises, and any male preoccupation with penis size is a response to that.
I don't see it.
It reads like another social stereotype
Some stereotypes are true.
Camille isn't some tumblr blogger, she's a respected scholar. She's studied society through the times and knows what she's talking about, women DO tend to be more judgmental and envious of each other and there are sociocultural reasons for that.
Black women too, some of whom are very rich 
I would argue it mostly comes from women and their preference for large penises, and any male preoccupation with penis size is a response to that.