Although they share the same basic concept, Twilight's vampires differ quite a lot from traditional vampire descriptions. Most vampire media today takes inspiration from the classic 1897 novel by Bram Stoker called Dracula. It's a name every vampire fan has heard of, raising the question of whether the vampire king exists in the Twilight universe. This may sound like a simple question, but the answer is far from clear, as the existence of Dracula raises far more questions than answers.
Before looking into whether Dracula lives as a real person in Twilight, it's important to know if people are aware of the character. Perhaps the strongest evidence to suggest that people know of Dracula is in Twilight: Breaking Dawn Part One. Early on, Edward recalls a time in the 1930s when he attended a screening of the movie Bride of Frankenstein. The Frankenstein movie's existence in the Twilight universe also suggests the presence of the Universal Monsters. This franchise consists of a number of horror films produced by Universal Pictures and features creatures such as Frankenstein's monsters, the Mummy and, most notably, Dracula.
Also supporting the Dracula movie's existence is the fact that Bella is aware of vampires before meeting Edward. She guesses that Edward is a creature of the night but doesn't believe in their existence until that moment. This suggests that the concept of vampires remains among humans and that there must be fictional portrayals of them for humans to not believe they are real. And with Dracula being arguably the most well-known fictional vampire, it would be strange for him not to be around in Twilight.
So assuming Dracula exists in the Twilight universe in film, it then begs the question as to whether the vampire king is real. A strong argument for why a real Dracula doesn't exist is because his powers and how he's described are very different from Twilight's vampires. For example, Twilight vampires have noticeable sparkling skin in the sun. So unless Dracula is some different kind of breed, it's unlikely he's alive as pop culture knows him. However, that doesn't necessarily mean a famous vampire called Dracula isn't around; it would just mean his book and movie depiction are not entirely accurate.
If the original Dracula novel exists in Twilight, it's also possible that the book was used as a way to throw humans off the scent of real vampires rather than being a coincidence. If vampires actively created books and movies to produce a false description of their kind, not only would they be harder to find, but humanity would take the idea of vampires much less seriously -- hence why Bella knows about vampires prior to meeting Edward but doubts their existence.
Either way, it's an interesting concept that Twilight has yet to answer clearly. But if it were to be addressed, there's a lot of potential history about the Twilight universe that could be expanded upon.