It's practically a given that all TV characters from Earth Prime-Time have doppelgangers in the countless alternate dimensions. This was illustrated time and again by 'Sliders' and 'Star Trek: Deep Space Nine'.
But some TV characters also have counterparts in other universes altogether. And by universes, I mean those created from the spark of Mankind's imagination, like Television. These include the worlds for theatre, film, novels, opera, myths & legends, song lyrics... probably even a universe for limericks as well.
(If there is one, Nantucket would be the center of that universe!)
Some of these characters are exactly the same, no matter in which universe they're found. For instance, a good example would be Cecily and Gwendolyn Pigeon of 'The Odd Couple', who are the same from the first Broadway production through the movie and into the classic sitcom.
The medical personnel of 'M*A*S*H' exist in three different universes - the worlds of novels, movies, and Television. Only Corporal Walter "Radar" O'Reilly has an exact duplicate in two of those universes. (I suppose in a way he could look like Gary Burghoff in the "Book World", if the person reading the Richard Hooker novel has him in mind...)
Tie-in novels based on TV shows must have their own little corner of that "Book World", in which their characters would look exactly like their counterparts who serve as their inspiration. In particular I'm thinking of 'Star Trek' and 'Doctor Who', but also of the 'Monk' and 'Diagnosis Murder' novels by Lee Goldberg, as well as countless other series tie-ins like 'Bonanza', 'The Prisoner', and 'Murder, She Wrote'.