If your system uses systemd, it has an etc/machine-id, which is used for a lot of different things. And changing it will break a lot of stuff, probably until you reboot. I guess you could write something to randomly shuffle it every time you reboot? But it is the go-to way for lots of programs (including browsers) to identify themselves. Which means (unless you have done the work to scramble your machine ID) you can be tracked on Linux as well.