Many of Sid James' characters were named Sid or Sidney, although this also happened from time to time with Kenneth Williams, Charles Hawtrey, Joan Sims, Barbara Windsor, and various others.
Carry On Camping probably contains the most uses of this trope, with characters including Sid Boggle (James), Kenneth Soaper (Williams), Charlie Muggins (Hawtrey), Bernie Lugg (Bresslaw), Joan Fussey (Sims), and Babs (Windsor).
The character of Sidney Bung in Carry On Screaming! was intended to invoke this, but when Sid James proved unavailable, the role was recast with Harry H. Corbett.
Dawson Casting: Frequently used. For example, in Carry On Sergeant, the National Service recruits, who should be in their late teens or early 20s, were all played by actors who were over 30 at the time of filming (in fact, Charles Hawtrey was over 40, Kenneth Connor nearly so). Similarly, in Carry On Camping, the finishing school girls were played by actresses in their late 20s or early 30s rather than in their late teens or early 20s (for example, Barbara Windsor was 31).
Doing It for the Art: Bernard Bresslaw in Carry On Up the Jungle went to the trouble of translating his lines into the appropriate African dialect. The rest said he could have just made up some Foreign Sounding Gibberish and no-one would be the wiser, while the extras playing the other Africans were of Caribbean origin and didn't understand him anyway. Years later, an immigrant from that area told him he remembered that scene and thanked him! (Sid James, who had lived in South Africa until his early 30s, also congratulated Bresslaw on his diligence.)
Fake Nationality: The primary recurring cast members were all British (except for Sid James, who was born and raised in South Africa but tended to play Cockneys, making him a Fake Brit), but many of them played a variety of other nationalities across the series, usually without affecting the relevant accent. Carry On Cowboy was a notable exception, with the entire cast affecting American accents (of varying quality) except for Jim Dale (whose character was British) and Charles Hawtrey (the incongruity of an Indian chief looking and talking like Hawtrey being a source of humour). Bernard Bresslaw combined this with Blackface or Brownface in several films, playing a Native American in Cowboy, an Arab in Follow That Camel, an Indian in Up the Khyber, and a sub-Saharan African in Up the Jungle.
In a similar vein, Wilfred Mott apparently served in the British Navy in the days of Napoleon (Carry on Jack) and later in the British Secret Service (Carry On Spying) before joining the Doctor in the TARDIS.
Frank Thornton, best known as Captain Peacock in Are You Being Served?, appears in Carry On Screaming as, appropriately, the owner of a clothing shop.
The Pete Best: If anyone qualifies for this title it would probably be Shirley Eaton, who was the female lead in several of the early films, but ceased to be a regular cast member just before the series really hit its stride. Additionally, Bob Monkhouse was probably the most famous cast member in Sergeant (aside from William Hartnell), but never appeared in any of the following films.
Prop Recycling: Carry On Cleo used the abandoned sets from Cleopatra when the latter production moved from Pinewood Studios to Rome.