I usually avoid delving into non-football topics, but lets address this.
I did cite the cultural example to drive home the point that there isn't any one universal set of mannerisms that one must abide by. It doesn't depend solely on your religion but also culture, social upbringing etc.
As for the religious angle, I'm not sure if you are speaking based on your time living in Islamic countries, your personal research or based on what's perpetuated by the western zionist media. But its a bit rich trying to educate a muslim woman who's lived in an Islamic country and extensively visited the others, about the virtues and vices about her religion. Both men and women are equal in value but they are not the same. Islam recognizes the critical differences and adjudicates accordingly. Women are given certain privileges that men dont have and vice versa, and there is a rationale behind everything.
Women are perceived precious members of society and mother granted the highest honour. Men are their protectors and providers. If men dont shake hands of women (outside of immediate family), its precisely out of respect for these women, not for the lack of it. These women are seen not just as mere individuals but as daughters, mothers, sisters of someone else. It has nothing to do with discrimination or inequality. From my lived experiences, I can tell you that I can walk out any time of the night and feel safe in an Islamic country than I have ever felt during my time in the west. I have also never been disrespected or felt deprived due to my religion. And countless other women resonate with this reality. That doesn't mean utopia or perfection ofc. The Islamic world is marred by extremism, economic unrest, political corruption, bad governance (party due to conflicts, war and partly due to incompetence), where religion is used a tool of diversion and power grab. Its not the religion per se that is the problem but rather its lack of meaningful and proper implementation in society.
Going back to this one incident, if Khabib didn't greet Kate with a handshake, its out of respect (instead he keeps his hand on his heart). Yes, its both a cultural and religious thing for him. I'd wager if a woman was in trouble he'd be the first in line to defend her before the majority of internet warriors could even contemplate. The only athlete who gave up his career at his peak to honour his word he gave his mother, amongst the few athletes who exercised restraint when his biggest adversary (of the west) was hurling abuses at his family including his deceased father. Khabib is a class act. Just because you have been conditioned to a certain way of life or manners doesn't mean there is no merit to what you deem as 'backward'.
My advice for you is try and expand your horizons with the sincere intention to learn, and you will end up appreciating your fellow humans and their way of life better.