As a Dark Lord of the Sith, Dooku was typically addressed as “Count” rather than as “Darth” in Dooku, who served as Qui-Gon Jinn’s master.
Count Dooku was introduced into the Star Wars canon in Attack of the Clones and his story has since been explored across a variety of media, especially in Cavan Scott’s book, Dooku: Jedi Lost. While the Jedi in AOTC refer to him as Count Dooku, Jango Fett mentions that he “was recruited by a man named Tyranus” to serve as the model for the Galactic Republic’s Clone Army. Indeed, the name bestowed upon Dooku by Darth Sidious, his Sith master, was Darth Tyranus.
Christopher Lee, who is particularly famous for his numerous performances as Count Dracula, undoubtedly influenced the decision to make his Star Wars character a Count as well thanks to his prominent back catalog. Furthermore, in-universe, Dooku is also Darth Tyranus, much like Anakin Skywalker eventually became Darth Vader. Sith bear their birth name as well as the name given to them by their master. The Jedi typically refer to Dooku as “Count” because that is the name they prefer to associate him with before he fell to the Dark Side. This is why, despite bearing both names, Dooku is most often described as a Count in Star Wars.
Many times there have been instances of out-of-universe preferences influencing decisions made in-universe within the Star Wars franchise. Samuel L. Jackson wanting his character to have a purple lightsaber is a classic example of this and since then, there have been some theories as to why Mace Windu wields such a lightsaber in the canon. With that said, Christopher Lee’s prolific Dracula performances in movies was most certainly inspiration for giving his Star Wars character, Dooku, a backstory that included him descending from a Count and Countess, therefore making him a Count of the planet Serenno.
Dooku had been a Jedi, a padawan to Yoda even, before he allied himself with Darth Sidious, working to bring on more star systems as Separatists to help further his master’s cause. Sidious may have given Dooku the title of Darth Tyranus, but because the Jedi Order knew him as Count Dooku, greatly preferring to not indulge his role as a Sith, they continued to call him by that name. Count Dooku trained many Jedi and Sith over his lifetime, and his various names would have been used accordingly depending on who was addressing him.
With an expanding canon world to help to explain some of these curiosities throughout the Star Wars universe, an in-universe explanation now exists to explain Dooku’s title of Count. Regardless, the nod toward Christopher Lee and his impressive career is a commendable one. Whether as Count or Darth, Dooku also led a prolific life, one that stretched across a variety of roles as a Count, Jedi, and Sith.