GNU Bash (4.2 as of this post)’s manual page says that its pattern matching operator =~
value =~ pattern
uses the ERE (Extended Regular Expression syntax) specification used by regex(3) which is the POSIX 1003.2 regular expressions format. Also, =~ has the same precedence as the == and != operators.
This is documented in the SHELL GRAMMAR section – Compound Commands – [[ expression ]].