The only one you really need to care about (especially under Linux) is PCRE,
Well, no. sed, grep, awk, vi etc. use POSIX regexes. GNU implementations also provide perl compatible mode via an unportable option. In modern programming languages like go and rust standard regex engines are compatible to RE2 - relatively new dialect developed in Google that is not described in the Friedl’s book (you may think of it as an extension of extended POSIX dialect). Even raku has its own dialect incompatible to perl as well as other ones.
Nowadays it is common to move away from perl-like engines, however they are still widely used in PCRE based software and software written in python, JS etc.