In casuals, it doesn’t really matter.
In tourney, sometimes I shake the other guys hand before the match and say “good luck” or something. It’s just a friendly gesture.
After the tourney, win or lose, I’ll shake your hand again, because it’s a friendly gesture.
Tourney’s aren’t just about video games. To me, tournies are half about the games, and half about the people. I like to make friends, especially with those in the FG community, so if you refuse to shake hands for really, no reason, you’re only portraying yourself as unfriendly. And of course, you can do fist bump/say GGs, etc. You just have to have some sort of sign of well, a friendly gesture. If you don’t, people will think “wow, that guy is a douchebag” before they even know you. First impressions are important, and if you don’t even try to be friendly, then you’re not gonna make friends.
So you might say “So I don’t want to make friends.” Fine, of course nobody is asking you to, but is it really fun to go to a tournament and have nobody to talk to, and have not a single person there respect you regardless of your skills? Imagine playing a tournament match against a guy who has a lot of friends. His entire crew is cheering him on. You have nobody cheering you on. Is that really fun? I’ve made some of my closest friends in the fighting game community. Yeah, some people are sore losers. I’m a sore loser, too, sometimes. But you still can’t take that away from your opponent in the tourney match. Your opponent wants to win, too. Nobody plays to lose. Not shaking their hands for trying to win is a dick move. So even if that’s not your intention, that’s what it’s seen as. If you get up and just walk away without saying anything or acknowledging your opponent or something, you’re not gonna have people wanting to talk to you or hang out with you. However, if you get up and say “I don’t do handshakes, but good games, bro” and walk away, people won’t hold that against you.
So it doesn’t really matter to other people. It’s really just about how you want to portray yourself.