There was a local commercial in Toronto about ten years ago for a garment store called "Tip Top Tailors". It featured a rather silly montage of a James Bond-type character saving a girl in a Thriller environment. At the end the man says, "Tailors. Tip Top Tailors".
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In the first episode of Full Metal Panic!, Sōsuke introduces himself to a hysterical runaway as "Sagara... Sōsuke Sagara," as he rescues her from the battlefield.
In Japanese, family names come first; Sōsuke's line in the Japanese dialogue is "Sagara... Sagara Sōsuke," which is more comparable to saying "James... James Bond," in English.
Heroman features an example without a first name: "Minami, Doctor Minami."
In Escaflowne, Hitomi introduces herself to Allen with her first name first, then her name in Japanese order (so it comes out as "Hitomi, Kanzaki Hitomi" in Japanese). In the English dub, she introduces herself as "Hitomi, Hitomi Kanzaki". Allen does not make any mistakes with her name, despite the fact that Asturian name order follows the English pattern.
Also played with in the Duran Duran video for the theme to A View to a Kill; toward the end of the video, the band's lead singer identifies himself as "Bon. Simon LeBon."
"Trench, Sylvia Trench" from Dr. No is actually the first charater to use this trope. It's in response to this introduction that Bond first utters the deathless line. And in the reboot Casino Royale it's "Mathis - Rene Mathis."
Goldfinger - Bond uses the line on Jill Masterson, then later on uses it on her sister Tilly, who impatiently cuts him off halfway through.
In Neverwhere, the protagonist fumbles over his initial self-introduction. Door doesn't seem all that surprised at meeting a man seemingly named "Richard-Richard-Mayhew-Dick".
Every single volume in the Geronimo Stilton series starts off with Geronimo introducing himself to the readers as "Stilton, Geronimo Stilton." One of the books was even titled "My Name is Stilton, Geronimo Stilton."
The aforementioned Sally Lockhart example, from the television adaptations of the novels by Philip Pullman.
In Mobile, Michael Kitchen's character introduces himself by saying "The name's West, David West". Kitchen played Bill Tanner in two James Bond films.
In Doctor Who, when the Doctor signs on with UNIT at the end of "Spearhead from Space", the Brigadier points out that he doesn't even know the Doctor's name, which the Doctor then gives as "Smith, Doctor John Smith."
In The Time Monster, Jo Grant is introduced to the Atlanteans as "Jo, Jo Grant". They proceed to call her "Jojo Grant".
In Dalek, when asked her name, Godard says "Godard, sir, Diana Godard."
Lampshaded in "The Wedding of River Song", where Amy bursts into the room with a gun and looking every bit like a secret agent and identifies herself as "Pond, Amelia Pond".
Played for laughs in an episode of Full House, in which a character who had lived in the Tanner's house many years ago consistently refers to himself as "Bond, Lou Bond," even pointing out his initials scratched in an attic wall: "B., L. B."
Chuck did this once or twice, but usually with an alias, not his real name.
LOST has this in spades. Nearly all -if not all- characters introduce themselves at least once -if not multiple times- in this fashion.
Torchwood: Captain Jack Harkness replies to an introduction of "Jones, Ianto Jones" with "Nice to meet you, Jones Ianto Jones".
On the border of this trope: in Star Trek: TNG, "The Outrageous Okona", Wesley tells Okona his name is "Wesley. Wesley Crusher," and also mentions he's an acting ensign. Okona says, in a wacky voice, "I'm glad to meet you, Acting Ensign Wesley Wesley Crusher."
And played straight in the DS9Bond-pastiche "Our Man Bashir" which starts with a scene where a thug is thrown through a window by an unidentified man in a white tuxedo, who proceeds to defeat another bad guy (seen in a mirror, no less) with the cork from a champagne bottle. The saved Damsel in Distress prompts with a "Thank you, Mr. —" at which point the camera finally swings around to reveal that it is Dr. Bashir (in a holosuit-program) who responds with "Bashir, Julien Bashir".
A variant occurs in The Russians Are Coming The Russians Are Coming. Walt Whitaker introduces himself to the leader of the shore party this way, but gets cut off after he says "Whitaker, Walt—". From then on, the officer refers to him as "Whitaker Walt".
The opening credits of the original half-hour version of Danger Man have Patrick McGoohan saying "My name is Drake. John Drake." This was 3 years before Sean Connery ever said "My name is Bond. James Bond."
This happened on an early episode of Green Acres when Oliver was introducing himself to some of the Hooterville residents, including Uncle Joe Bradley (crossing over from Petticoat Junction).
Oliver: Douglas. Oliver Wendell Douglas. Uncle Joe: You've got enough names for two people.
The TV version of Starman had the Starman (Who took on the body of Paul Edward Forrester) briefly address his son as "Scott Scott Hayden" despite him knowing that "Hayden" was Jenny's surname.
In an episode of Charlie's Angels, Bosley introduces himself this way, only to get the "Hello Bosley Tom Bosley" response. He tried to correct it but ended up doing it again.
In a flashback in Smallville, Ma Kent tells the sheriff that the kid is called Clark "... because we felt my family's name would be appropriate..." (actually they hadn't thought of any and that was the first she made up). Dunno if that's canon in other incarnations, but at least in Smallville, the Man of Steel can rightfully present himself as "Clark Kent Clark".
Sterling does it in the Leverage episode "The Zanzibar Marketplace Job". He's playing hero in front of a bunch of news cameras, so it actually makes sense.
In the first scene of the first episode of Party Down, a character gives the setup: "That is quite a motto, Mr..." The response: "Donald. Ron Donald."
When Booth and Brennan go to the UK in Bones, Booth is charmed that the gun he's issued is the same kind Bond used, and proceeds to answer the phone this way.
In The A-Team's Season 5 episode, "The Spy Who Mugged Me", which is an Affectionate Parody of James Bond films, Murdock is sent undercover as a spy named Logan Ross. Naturally, knowing what kind of story he's in, Murdock spends the rest of the episode introducing himself as "Ross, Logan Ross" with a very good impression of Sean Connery's voice. He also goes to the bar and orders "orange juice: shaken, not stirred."
Parodied in the 80's Alfred Hitchcock Presents episode "Diamonds Aren't Forever," in which the main character, a spy with the first name James, repeatedly tries to do this trope, but whenever he or another character is about to say his last name, they're always interrupted somehow.
Night Court once had Dan introduce himself as "Dan. Dan Fielding" to which the Eskimo he was talking to thought his name was Dandan Fielding.
During a fantasy sequence on The Fresh Prince of Bel-Air, Carlton introduced himself like this while saving a young woman from a mugger.
In the Gorillaz biography Rise of the Ogre, 2D is quoted as explaining: "I know there's a rumour going round that my real name is Stuart Tusspot, but that's not true. It's Pot. Stuart Pot." (It was originally Tusspot, but his father had it legally changed for obvious reasons.)
The first song on The Mars Volta's album Frances the Mute (second if you count the title track, which was intended to lead off the album but had to be left off due to Executive Meddling) is entitled "Cygnus....Vismund Cygnus".
In The Curse Of Monkey Island, Guybrush can introduce himself to King André this way, either under his own name or other aliases; but no matter which other aliases he chooses, André will know Guybrush's name anyway by calling him "Mister Threepwood".
Linkara: It's amazing how adding just that one word drains the response of any kind of power.
Parodied in an episode of Disney's Hercules that spoof James Bond: Hercules introduced himself as "Les... Hercules" (and he was called for the rest of the episode as Mr. Les).
In the Bond parody on Pinky And The Brain, "To Russia with Lab Mice", Brain introduces himself to Russian mouse Mousey Galore as "Brain... The Brain."
On an episode of Arthur, a movie preview is shown for an entry in the James Hound series. In it, the hero introduces himself as "Hound. James Hound." Arthur does the same later when imagining himself in the role.
In an episode of Ben 10 Alien Force Ben introduces himself to a Highbreed who's his partner in an Enemy Mine as "Ben, Ben Tennyson". From that point on the Highbreed addresses him as "Benben Tennyson".
In an Episode of Timon & Pumbaa Timon introduces himself to the bad guy as Mon Timon For the rest of the episode the bad guys calls him Mr. Mon
Some of the promotional videos for Kim Possible involved the whole Bond "Gun Barrel Scene" with the Disney logo instead of the gun circle and with Kim's silhouette. She then invokes the trope: "Possible. Kim Possible."
There's also "Shtoppable. Ron Shtoppable." With a lisp that Sean Connery didn't pronounce that much until later in life.
Another obvious parody in Wakfu episode 19 (as well as a Punny Name in French): "My name is Monde... Smisse Monde."
In one episode of Jackie Chan Adventures, Jackie introduces himself to an art collector as "Chan, Jackie Chan". For the rest of the episode, the guy calls him "Chan Jackie Chan".