The same is absolutely not true for boys. Just look at what happens to a little boy if he decides he wants to wear girls' clothes or play with dolls. Whereas a girl wearing pants and playing with trucks will, at worst, be called a tomboy.
How much of that is due to the efforts of men or boys? Or how much is due to bullying by other girls in middle school and high school?
As far as I can tell, schools have put in enormous effort to stop boys from fighting each other. I can’t find much evidence of their efforts to stop girls from bullying each other by spreading rumours. Girls are highly susceptible to peer pressure, particularly from other girls.
Just a few quotes from /r/Parenting:
"BUT his dad had a conniption when he found out that I bought him a baby doll and he thinks that he should not play with “girl” toys." https://www.reddit.com/r/Parenting/comments/b403o6/how_do_yo...
"A lot of adults (men, women, young, old) make snarky comments in my son's presence about his toy selections (doll houses, play kitchen sets)." https://www.reddit.com/r/Parenting/comments/b403o6/how_do_yo...
"Now, my husband generally doesn't mind that my step-son plays Barbies or dollies with his sister. However, he draws the line at actually buying "girl" toys for him." https://www.reddit.com/r/Parenting/comments/53p1ut/boys_and_...
"I'm constantly reminded my girl isn't very "girly". At times I'm even asked whatever I'm concerned about her enjoy so many "boy's things"" https://www.reddit.com/r/Parenting/comments/2ke89r/parents_w...
[edit] what I mean is simply that those kinds of sentiments might be a reaction to perceived bias in culture, expressed as bias but not driven by same, if that makes sense. A kind of re-enforcing meta-bias. Again, doesn't make the effect any better.
The key is to acknowledge first that our society is incredibly gendered (and racist) in ways we often oblivious to.