That's a rather shallow interpretation of Zoro if I'm being honest. Zoro said he wants to be the best swordsman there ever was, which possibly mean there were better swordsmen than Mihawk before his era, but obviously Zoro can't fight the dead so he needs to go by some other metric to remove doubt of him being the best swordsman there ever was. Zoro also wants his name to reach the heavens, which implies he'll do not great, but fantastical things to have his name be amongst the greatest in the world.
Zoro's dream thus far has been rather flat, and many things that surround the lore of swordsmanship is shrouded in mystery and hasn't really been lacking in application such as grade blade, like do the different ranks between named swords even matter. Mihawk's himself really hasn't displayed what makes being the World's Strongest Swordsman impressive, which lessens the reputation of the title.
It also doesn't help that many of Zoro's enemies, especially in the Timeskip, aren't even swordsmen themselves, which makes it difficult to compare how well Zoro's improving in swordsmanship, not strictly power. And other possible swordsmen of the world may not even be swordsmen just like King, so that just murks the water of who Zoro can compare to other than Mihawk, who himself hasn't done much.
I understand that Zoro puts Luffy's dream over his own, but it's to a point that his own dream has been almost fully overshadowed by Luffy's dream, not given anyway to grow and be fruitful as opposed to Luffy's. Unlike Luffy's dream, Zoro's doesn't earn recognition whenever he spouts off about it, which kills any interesting interactions surrounding the title of "World's Strongest Swordsman".
Overall I feel that Zoro's dream is practically irrelevant in the One Piece world as only 2 people care about it, and Mihawk doesn't do much to give the title any flair.
Zoro's dream thus far has been rather flat, and many things that surround the lore of swordsmanship is shrouded in mystery and hasn't really been lacking in application such as grade blade, like do the different ranks between named swords even matter. Mihawk's himself really hasn't displayed what makes being the World's Strongest Swordsman impressive, which lessens the reputation of the title.
It also doesn't help that many of Zoro's enemies, especially in the Timeskip, aren't even swordsmen themselves, which makes it difficult to compare how well Zoro's improving in swordsmanship, not strictly power. And other possible swordsmen of the world may not even be swordsmen just like King, so that just murks the water of who Zoro can compare to other than Mihawk, who himself hasn't done much.
I understand that Zoro puts Luffy's dream over his own, but it's to a point that his own dream has been almost fully overshadowed by Luffy's dream, not given anyway to grow and be fruitful as opposed to Luffy's. Unlike Luffy's dream, Zoro's doesn't earn recognition whenever he spouts off about it, which kills any interesting interactions surrounding the title of "World's Strongest Swordsman".
Overall I feel that Zoro's dream is practically irrelevant in the One Piece world as only 2 people care about it, and Mihawk doesn't do much to give the title any flair.