Mihawk Backstory imo is complete Opposite of Zoro's, just like Big Mom's was Opposite of Sanji's,
For example, Big Mom was abandoned by her Family for being Too Strong while Sanji was also abandoned but for being Too Weak,
Big Mom met & bonded with Streusen who has Infinite Food because She ate too much, while Sanji met & bonded with Zeff who had No Food because He endured Hunger, or the fact that Big Mom lost Carmel while Sanji lost Sora, both Victims of turning Children into Soldiers, but Carmel died supporting that while Sora died for opposing it ... etc
In other words, it's like Same Coin (Similar Fate) but Different Sides (Opposite Reasons),
So in Zoro's Case, instead of being Abandoned by his Family, his Case was that He got Lost & couldn't go back to his Home Island, this means that Mihawk is also unable to get back to his Home but for Opposite Reason, in other words, it's his Island that is Lost, not him
The Rest of Story would follow same Pattern as well, meaning Mihawk too lost someone close like Kuina, He received Yoru similar to Wado but in Opposite Scenario to Zoro's, He then became Marine Hunter instead of Pirate Hunter ... etc
However, the most important thing is that Mihawk's Island is "Lost" imo, (Not Destroyed but Lost)
Zoro became Pirate Hunter because after being Lost, He had no choice but to be a Bounty Hunter to Survive,
So this explains why Mihawk became a Marine Hunter, it's because He was attacking Marine Bases to try & discover anything about his Island,
I only know Three Locations that People can't seem to find:
It's either Laugh Tale or Emerald City or Island of Dreams