I'm a dude and usually say "Can I please have..." but I'm from a Commonwealth country and I think there's a tendency for British/Commonwealth people to be less brash than Americans.
Can confirm. I'm Dutch, my GF is British. In Dutch, "Can I have x" is considered polite, but as it turns out it's quite rude in English, it needs a "please" somewhere.
Remarking on possible cultural differences between countries is in no way a rebuttal of the actual point. A rebuttal of the actual point would be "In my country, men and woman both speak the same" not "I see myself as more polite than you because of the cultural norms for my country being different from the cultural norms for your country."
I'm also from a Commonwealth country, and I've never detected a significant difference between the way people buy things at shops or order at restaurants.
Here (Australia) it's just rude people that say "Give me ..." and polite people who say "Can I please have ..." etc. - no clear split between male and female.
That's not to say your point is necessarily wrong, just that at least that particular example is highly region-specific.
A different example that hopefully will not result in some tangent about how rude Americans are or some nonsense:
I was a homemaker and mom for years. My sister had a serious career and delayed having a baby. When I was younger, I was sometimes weirded out by her framing. When I repeated things she said to other people, sometimes people actually remarked that she "spoke like a man" or similar.
I think there are differences in how men and women typically express themselves. Some of this may be rooted in different experiences that aren't per se gender specific -- ie men are more likely to be in charge and being in charge shapes language. Then men being in charge means men are more likely to emulate "boss" language. It becomes masculine by association.
But I think that a difference does exist in the aggregate.