Why does people complicate things?
Mihawk went on a Journey to become Greatest Swordsman, he became Super Famous even during Roger's Era
And one day he met Shanks & deemed him as the Best & it's obvious he saw his CoC
They had Epic Duels that were the talk of entire Grand Line, where even Strongest Man couldn't forget them & deemed them Legendary
Mihawk became known as Strongest Swordsman & most World Swordsmen loved the idea of meeting him & facing him one day
After some time, Shanks lost his Hand for future Generation & Mihawk lost interest in fighting him anymore cuz he is handicapped.
Shanks didn't let that loss effect him, & he aimed to conquer part of New World & become an Emperor
It's not like Shanks wasn't in the progress of becoming Yonko when he was Fighting Mihawk, because WB himself that everyone was shocked when Shanks returned with One Arm, that's because he was already a Huge deal in New World & sooner or later, everyone expected him to be considered a New Emperor
Mihawk is Strongest & Shanks is a Successful Pirate Captain (Actually Hyped by Oda to be The Best)
Nothing says Shanks surpassed Mihawk, nothing says he wasn't Strong enough to become Yonko back then, nothing says he changed his Fighting Style ... etc. These are just Fan Headcanons