I'll mention this once, and hopefully someone that understands my words will remember it if this ever comes up again.
Jordan is considered the greatest basketball player, before him that title belonged to Karim Abdul, someday that title will belong to most likely Lebron.
Is Lebron the best player? To be honest I'd prefer to have Carmelo Anthony or Tracy McGrady on my team rather than Lebron, why? Because they can do anything he can do and more. The difference is image, Lebron has the image of current greatest young star, and that will likely continue no matter how many more titles Wade wins.
Mihawk works for the Marines, the Marines are policing the world, just as most governments claim to their people "we are the greatest nation in the world!" and would never admit that a player/scientist or other political system is superior, the marines will never as well.
Who is the best swordsman in the marines? Mihawk, does that mean he cannot be defeated by another swordsman? No, the marines want the world to know their force is above the rest of the world, so they claim to Mihawks fame as the greatest. To me that means that Shanks can and would defeat Mihawk easily, but Mihawk can defeat Shanks at least 50% of the time, and the image of Mihawk being stronger is profitable to the marines, and to the mindset of the people who don't want to have to worry if Mihawk is ever facing another enemy. It is beneficial for the people to feel safe and not worry about even the possibility of one of the marines officers losing (since it is the Marines and even Shichibukai that protect them).
This to me, is why Mihawk has his claim to fame, nothing more nothing less. It's not that he's not the greatest, but it's not that no one else that uses a sword can defeat him either. This is also why Zoro wants to defeat Mihawk and not Shanks, to his small village and Dojo, his sensei had only heard of Mihawk as "the greatest" he probably hadn't heard of Shanks, or of Kaku or of any other swordsman, but just like people that know NOTHING about basketball know the name Jordan, he knew the name Mihawk.