The POSIX shell does
not implement regex.
What is used both in case/esac and globbing are "shell patterns." They are also found in variable pattern removal with ${X% and ${X#.
In "The Unix Programming Environment," Kernighan and Pike apologized for these close concepts that are easily mistaken for one another.
"Regular expressions are specified by giving special meaning to certain characters, just like the asterix, etc., used by the shell. There are a few more metacharacters, and, regrettably, differences in meanings." (page 102)
Bash does implement both patterns and regex, which means discerning their difference becomes even more critical. The POSIX shell is easier in memory for this reason, and others.
http://files.catwell.info/misc/mirror/