Mihawk is the World's Strongest Swordsman currently. As it is Zoro's, a main characters, dream to surpass him, then it makes absolutely no narrative sense that it is a "fake title", or whatever gibberish people like to say about him.
If someone (and it seems to be Shiryu you're focusing on) comes along and beats him, then yes, he is the World's Strongest Swordsman.
However
Zoro's dream is still narratively, the most important thing. Just because Shiryu (in this scenario) could beat Mihawk, that does not mean everyone else could, and his title was false. No more than Zoro beating Mihawk would suddenly make Mihawk's title worthless. It just means that there's another character, ahead of Zoro, who's better than every other swordsman, and it then becomes that guy Zoro has to surpass.