I don't know the exact significance of the word "Darth" in the Sith world, but I think it was a word of honor for powerful Sith. That's why Anakin was given this word in his new name.
All powerful Sith in canon had this word in their names: Darth Plagueis, Darth Bane, Darth Sidious, Darth Vader, Darth Maul, etc.. As such name similarity can't be names given by their parents, I can assume they were given such words in names as a honor.
Now, look at Count Dooku.. he was a powerful Sith too. So, why wasn't he given this word of honor?