Here’s how the nickname came about, as described on p. 71 of All the President’s Men:
Woodward had a source in the Executive Branch who had access to information at CRP [Committee to Re-elect the President, known as ‘Creep’] as well as at the White House. His identity was unknown to anyone else. He could be contacted only on very important occasions. Woodward had promised he would never identify him or his position to anyone. Further, he had agreed never to quote the man, even as an anonymous source. Their discussions would be only to confirm information that had been obtained elsewhere and to add some perspective.
In newspaper terminology, this meant the discussions were on “deep background.” Woodward explained the arrangement to managing editor Howard Simons one day. He had taken to calling the source “my friend,” but Simons dubbed him “Deep Throat,” the title of a celebrated pornographic movie. The name stuck.
Felt did not choose, and presumably knew nothing of, the nickname he'd been given by Post editors.
In 1972, newsrooms were still places of traditional masculine joking, and the nickname wouldn’t have seemed especially shocking. The phrase—if not its literal meaning—had entered the popular lexicon thanks to newspaper advertising and talk-show discussion of the movie. Interestingly, the book also recounts that on Feb. 26, 1973, as CRP was trying to serve subpoenas to several reporters, Managing Editor Ben Bradlee wanted Bernstein out of the building and told him to “go see a movie.” Bernstein went to see the movie “Deep Throat.” ATPM p. 260.