Most of the Arrancar seem to have some sort of honor hangup where they need to know their opponent's name. Byakuya flat out refuses to give his because he doesn't see any point in giving his name to someone who's about to die, and Mayuri only asks Szayel's name because he wants to know what to label the jar he'll be in. Yes, all of them are guilty of adding in their rank for no real reason. Only Ulquiorra and Yammy were late on it; Ulquiorra didn't bother with his last name until he'd already killed Ichigo, and Yammy did it when he off-screened Rukia, Renji, and Chad.
This trope is directly invoked by Ikkaku, who believes that you should only introduce yourself to an opponent if you intend to kill them. Which makes this line all the more badass.
"I'm the 3rd seated officer of Zaraki's division, Madarame Ikakku! You don't have to tell me your name, you just have to remember mine...because it's the name of the man who'll kick your ass!"
Zigzagged with the Vandenreich, as its members have varying degrees of Combat Pragmatism. The Stern Ritter who fights Kyouraku during the first assault clearly feels no need to introduce himself, as 40 chapters later we STILL don't know his name. On the other hand, Haschwalth, Kyouraku's opponent in the second assault, makes a very formal introduction, which Kyouraku reciprocates.
"I, Misao Makimachi, see no compelling reason to give villians like you my name!"
Dragon Ball Z (technically Dragon Ball Kai): "I'm a proud Saiyan who calls Earth home, and I'm here to defeat you. I am the warrior you've heard of in legends, pure of heart and awakened by fury. That's what I am. I AM THE SUPER SAIYAN, SON GOKU!"
Fate/Zero: Rider particularly does this, in the form of exposing himself as Iskandar, the King of Conquerors.
Katekyo Hitman Reborn!: Hibarin from Monster Tamer Tsuna, "I am Hibarin, the dreadful vampire. My address is Demon Valley 2-4. I love Salisbury steak and coconut juice. My hobby is to have fun by myself. And to bite weak people to death. And to play with birds." Cue Tsuna lampshading this trope, "He's giving an introduction when nobody asked for one! And it's taking forever." And Hibari finishing his monologue with this: "And to patrol my castle. That's right. I have one last thing to say. I'm not much of a talker.
Light Yagami from Death Note informs Naomi Misora that he is Kira, approximately one second before her death kicks in.
Magical Girl Lyrical Nanoha A's: "Time-Space Administration Bureau Part-time mage, Fate Testarossa." She does this right after saving Nanoha and right before attempting to place Vita under arrest.
The heroes have a catchphrase that precisely announces a 'I AM/WE ARE XXX' shout by whoever it is to kick ass: WHO THE HELL DO YOU THINK I AM/WE ARE?!"
This trope became Nia's first Crowning Moment Of Awesome! "The Spiral King Lordgenome's eldest daughter, Nia Teppelin commands you to stand down, Andiane the Elegant! Are you aware of who I am?" Simon and Yoko instantly did a "holy shit!" expression upon hearing it.
Simon: My Name is Simon. Leader of Team Dai-Gurren. If you've decided to be a wall that's standing in my way, I've got something that will open a hole in you every time. And that something is MY DRILL!
In A Certain Magical Index, magicians have a Magic Name, which they typically only reveal in a very serious situation, like when they declare they will kill someone, or when they need to use their full power.
"That's it! Don't you dunderheads know who I am? I'm Toph Bei Fong, the greatest earthbender of all time! When my friend tells you to calm down, you calm down!"
This classic from 'The Mighty Thor' Vol 1, #379 in 1987, by Thor himself to Jormungand.
But I have many names, serpent, even as you. Vingthor the Hurler, Longbeard's Son, have I been called. Hrodr's foeman, too. In Tyr's ancestral home, wisest Hymir knew my name as Veur; unhappy Hrungnir's playmate some have called me. East of Elvigar in Gianthome they whisper Hloriddi's name. My father called me son. My mother called me darling. And beneath the vaults of Heaven, I am Thor Odinson, the Thunderer, Jormungand's Fear!
Night Raven: Who do you think you are, you wimp? Fluttershy: My name is Fluttershy! I’m a member of the Mirage squadron! And I've sworn to protect Equestria and my friends! I’m a pony that represents the Element of Kindness! But, to you... I will show no mercy!
This is after he quotes the 'to the pain' speech to the Mouth of Sauron.
In the Horseshoes and Hand Grenades sidestory, Month of Sundays: "I am Akiko Narumi-Terui, daughter of the famous Souichi Narumi... and the new Kamen Rider Skull! Prepare to die!"
In the fan film The Jedi Hunter, which spoofs the Star Wars franchise in the form of The Crocodile Hunter with Boba Fett in the Steve Irwin role, Boba utters the trope almost word-for-word in his climactic showdown with Mace Windu:
Boba Fett: G'day, mate! My name is Boba Fett. You killed my father. Prepare to die.
In The Godfather Part II, Young Vito Corleone/Andolini confronts an elderly Don Ciccio, who asks "Tell me, who was your father?" Vito replies "My father was Antonio Andolini, and this is for you!" He then proceeds to stab him and start a gunfight.
Apocalypto: "I am Jaguar Paw, Son of Flint Sky. My father hunted this forest before me. My name is Jaguar Paw, I am a hunter. This is my forest, and my sons will hunt it with their sons after I am gone. Come on! COME ON!!"
John Harrison was a fiction created the moment I was awoken by your Admiral Marcus to help him advance his cause, a smokescreen to conceal my true identity. My name is... KHAN.
In The Princess Bride all the way through the book Inigo has been practicing his speech. After a huge buildup, when he finally gets to deliver it in the climax to his lifelong search, honing his sword skills for the battle of his life, his enemy runs away like a little girl. Of course, Inigo then chased Rugen down, and then repeated the speech several times with increasing volume as they dueled.
Original Conan the Barbarian loved this. Best known example is probably in The Frost Giant's Daughter, where Conan and his opponent exchange this.
"Man," said he, "tell me your name, so that my brothers in Vanaheim may know who was the last of Wulfhere's band to fall before the sword of Heimdul."
Used to great effect in the Codex Alera book Captain's Fury, when Tavi reveals that he is Princeps Gaius Octavian. Happens to be punctuated by a purely coincidental, plot relevant, volcanic eruption hundreds of miles away.
In 1634: The Galileo Affair, by Eric Flint and Andrew Dennis, there's a line something like "My name is Ruy Sanchez de Casador y Ortiz. Prepare to die," then subverts the reference's Heroic Second Wind. Our friendly neighborhood old fart proceeds to slaughter and get cut up, while his much-younger girlfriend-from-the-future wonders when he had a chance to read The Princess Bride and another uptimer realizes he didn't — he's Inigo Montoya's prototype.
Strider: I am Aragorn son of Arathorn, and I am called Elessar, the Elfstone, Dunadan, the heir of Isildur Elendil's son of Gondor. Here is the Sword that was Broken and is forged again! Will you aid me or thwart me? Choose swiftly!
For greater Bad Ass-itude, he later calls out Big BadSauron via Palantír, saying much the same (except for the aid or thwart me bit).
"Who am I? I am Susan Ivanova. Commander. Daughter of Andrei and Sophie Ivanov. I am the right hand of vengeance, and the boot that is going to kick your sorry ass all the way back to Earth, sweetheart! I am Death Incarnate, and the last living thing that you are ever going to see. God sent me."
Doctor Who: The Doctor himself has done this on many occasions, especially post-Time War when he is the only living Time Lord. Some examples:
The Doctor: I'm the Doctor, and you're in the biggest library in the universe... look me up. ** "The Eleventh Hour":
The Doctor: Is this world protected? 'Cause you're not the first one to have come here — oh, there have been so many. And what you've got to ask is: what happened to them?... Hello. I'm the Doctor.Basically... Run.
NCIS Season 7, episode 1 does this with a slight twist. Towards the end, after McGee tries and fails to kill Saleem, Tony says to Saleem, "I told you about every member of the team, except myself. The part I play... I'm the wild card. I'm the guy who looks at the reality in front of him and refuses to accept it." (Technically Tony tells Saleem his name much earlier, but this is the point where he really tells Saleem WHO he is, and right before Saleem is killed.)
Done as strait as an arrow in season 8, episode 23 with Mike Franks' last stand.
Franks: My name is Mike Franks! I figure I got one more fight in me... You want it?
Seinfeld: In the episode where George does the opposite of what he normally does, he meets an attractive woman. Since George normally gives fake identities when meeting people, he ends up doing this. It actually works
George: Hi, my name is George. I'm unemployed and I live with my parents.
Nikita: Alex does this to Vlad, the man who sold her into white-slavery.
Alex: I am Alexandra Udinov, daughter of Nicholai Udinov. And this is your reward.
Mook Guard: You work, and you live. That is all. You do not complain or laugh or do anything besides work. Whatever you thought, whatever you were does not matter. You are no one now. You mean nothing. (to boy) Who are you?
Katie: I'm Katie fucking Fitch, who the fuck are you?
LOST: Season 5, episode 12, Ben Linus upon approaching Penny Widmore: "Hello. My name is Benjamin Linus". He wasn't prepared to duel her, though; He was about to kill her when Desmond, whom he just tried to neutralized by shooting him, attacked him from behind.
Mitchell from Being Human uses a non-violent version of the same style of threat to get a room full of Vampires to shut up and listen to him.
My name is John Mitchell! And I've killed more people than you've met.
Also, Johnny Cash at the start of every show he ever did:
"Hello. I'm Johnny Cash."
In The Distillers' "The Young Crazed Peeling", one of the lines in the first verse of the song goes like this:
My name is Brody, I'm from Melbourne
Fitzroy Melbourne, Fitzroy Melbourne
I grew up on Bell St. then on Bennett St.
My mom kicked out my dad for battery
Myths & Religion
In The Odyssey, Odysseus just kicked the butt out of a Cyclops, Polyphemos. As he leaves he brilliantly declares "Kyklops,/if ever mortal man inquire/how you were put to shame and blinded, tell him/Odysseus, raider of cities, took your eye:/Laertes' son, whose home's on Ithaka!" He may as well have given him the precise longitude and lattitude of his house, and just thrown in his social security number, too.
Protip: if you live in Greece's "Age of Heroes" and you're already well acquainted with several mean-tempered gods, wounding the son of Poseidon is almost as dumb as killing Inigo Montoya's dad.
Chrono Trigger has Frogs' heroic speech. "My name is Glenn! Long have I carried Cyrus's hopes and dreams, and now I bear the Masamune as well! Henceforth, I claim them as my own! I shall slay the Fiendlord Magus and restore our honor!"
Hakumen: I am the white void. I am the cold steel. I am the just sword. With blade in hand shall I reap the sins of this world, and cleanse it in the fires of destruction. I am Hakumen! The end has come!
Reznov: My name is Viktor Reznov. AND I WILL HAVE. MY. REVENGE!!''
Assassin in Fate/stay night introduces himself formally on initiating a duel, though he doesn't insist Saber identify herself.
In Super Robot Wars, anyone who's about to be on the receiving end of Sanger's BFS will at least get to know that "I am Sanger Zonvolt, the sword that cleaves evil!" Wodan Ymir does pretty much the same thing.
Sir Ano does this in one of the Oaklore quests in Dragon Fable.
Sir Ano: My name is Sir Ano de Berg'rac. You're planning to kidnap my princess. Prepare to die.
Ezio in Assassin's Creed II after he stabs Uberto Alberti to death: "The Auditore are not dead! I'm still here! ME! Ezio! Ezio Auditore!"
From Fire Emblem, we get KIERAN, LEADER OF THE FIFTH PLATOON OF THE CRIMEAN ROYAL KNIGHTS! / SECOND COMMANDER OF THE CRIMEAN ROYAL KNIGHTS!.
Dwarf Fortess does this in Adventure Mode with anyone you can talk to, especially the ones you've been sent to kill. Speeches typically have their name, relation to other histrorical figures, kill list, and then the obligatory "prepare to die" if they're trying to kill you.
Gruntz: when a grunt carries a sword and is forced to walk around non-stop for longer periods of time, you can hear him saying this: "My name is Grunt Montoya. You killed my father. Prepare to die!"
Inverted in Umineko no Naku Koro ni Episode 6; when Erika delivers her dramatic introduction to Crowning Music, the battle is already over, she's lost badly and is prepared to die. She uses a similar line as a straight example for an intro in Ougon Musou.
The GEKICOM Elementals and the STRUQ Team in Ilivais X do this like it's required by law. They're all Large Hams anyway. En Arteya did as well, and technically Ashe too. Justified in that otherwise we'd have no idea who they are, due to their Monster of the Week nature.
Also Katara in "The Southern Raiders", and her insistence that the soldier remember her and her mother.
Ben 10: Alien Force: Said by Kevin. "My name is Kevin Ethan Levin. You killed my father. Prepare to die." Too bad it didn't work.As usual. Though Kevin does eventually manage to kill the guy at the end of the episode.
Word of God says that Kevin's mom actually read The Princess Bride to him when he was little.