The Danza - 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.
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.