In this usage, the recurring personalities adopted by Laurel and Hardy through all of their many comedy films (although they often played characters with different names and occupations) would qualify as their shtick.
A comedian might maintain several different shticks of this sort, particularly if appearing in a variety show encouraging development of multiple characters, such as Saturday Night Live.
In common usage, the word shtick has also come to mean any talent, style, habit, or other eccentricity for which a person is particularly well known, even if not intended for comedic purposes.
For example, a person who is known locally for an ability to eat dozens of hot dogs quickly might say that it was his shtick.
For this reason, journalists and commentators often apply the word disparagingly to stock replies from politicians.