It annoys me when people claim this. It depends on the game, distro, proton version, what desktop environment, plus a lot of other things I have forgotten about.
Also latency is frequently worse on Linux. I play a lot of quick twitch games on Linux and Windows and while fps and frame times are generally in the same ballpark, latency is far higher.
Other problems is that proton compatibility is all over the place. Some of the games valve said were certified don't actually work well, mods can be problematic, and generally you end up faffing with custom launch options to get things working well.
Many of those games mysteriously fail to work for me, almost like Proton has a problem on my system in general and I am unable to figure it out. However, in the past I got games that are made for Windows to work better on WINE than on Windows. One of those games is Starcraft 2 when it came out. On Windows it would always freeze in one movie/sequence of the single player campaign, which made it actually unplayable on Windows, while after some trial and error, I managed to get a fully working game on GNU/Linux, and was able to finish the campaign.
This goes to show, that the experience with Proton and different hardware and whatever it is in system configuration is highly individual, but also, that games can indeed run better using WINE or Proton than on the system they were made for.
Consistency is better than any theoretical FPS improvements IMO.
Often for games that don't work with modern Windows there are fan patches/mods that fix these issues.
For games that are modern frequently have weird framerate issues that rarely happen on Windows. When I am playing a multiplayer, fast twitch game I don't want the framerate to randomly dip.
I was gaming exclusively on Linux from 2019 and gave up earlier this year. I wanted to play Red Alert 2 and trying to work out what to with Wine and all the other stuff was a PITA. It was all easy on Windows.
(I don’t even know you could actually start mainstream games on BSD or not)