Why you aren't, but should be shipping Mihawk and Perona together, an essay.
First of all, Perona and Mihawk are adults. When they met, Perona was 23 and Mihawk was 41. Yes, this is an eighteen year age gap, but I want to start with this because it's the primary reason I see for people not liking the ship. After two-three years living under the same roof, by the time Perona leaves to go find Moria she is 25-26 and he is 43-44 (depending on how long you believe is sailing time between arcs and when their birthdays may fall between the time Zoro was dropped off at Sabaody and the summit at Mariejois took place.
Let's pretend for a moment Mihawk really is a vampire. This means one of two things, either he died at 41 and is actually centuries old, or he became a vampire sometime in his 20s and is legit 44-45 years old. Either way, Mihawk comes off as extremely old fashioned in the way he talks, the way he dresses. I would say he, like Perona, is a gothic romantic. Romantics tend to put their other passions first in favor of actual romance, keeping this in mind this would indicate that like many handsome dark brooding autistic men before him (the good ol' Darcy type) he has put romance on the wayside because he wants a love that is real. This is not an uncommon romantic trope amongst gothic romantic and vampire lovers and generally stories with immortals.
Often in fiction and romance novels the centuries old immortal is paired with a mortal (usually completely human) woman between 18-35. This type of romantic man has waited centuries for the right person they could form a lasting bond with and has not had much experience with actual love, if at all. For a few examples: ACATOR series Feyre and Rhys got together when Feyre was 20, and before that she was even younger with her first faerie boyfriend. In All Souls Trilogy Diana is suggested to be in her late 20s/early 30s and Matthew is pretty much old as dirt. In the Twilight Series Bella was 17 when she met Edward, who was 104. These are a trope for a reason, and when done right it’s a pretty good trope.
I'm going to include an incredibly unpopular opinion here, but "Goth Fam" is weird and here's why. While amusing in theory, portraying two grown adults and another grown adult as children and one adult is incredibly infantilizing. I won't disagree Zoro and Perona have a very close relationship where Perona especially is seen taking care of and is incredibly protective of him, I don't see where this necessarily makes them sibling coded (the same way I don't think Nami and Zoro are sibling coded). I won't get into that in more detail here about my thoughts on Zoro and Perona, but here's my hot take.
Both Zoro and Perona are incredibly autistic coded, which makes drawing them drawn as children with Mihawk incredibly cringe. Mihawk did not know Zoro and Perona as anything but adults and does not get a right to claim either of them as his children unless they offer that up first. We never see either one of them directly call Mihawk a father figure.
A 5’3, petite queen, the way Perona dresses and wears her hair is a fashion statement. I don't think she would like being portrayed as an actual child. Post-time skip she even has a tattoo on her arm the same place Nami’s tattoo is placed on her left arm, permanently branding herself and displaying her loyalty as a Thriller Bark Pirate. Perona does not need another father figure, she has not only Moria, but multiple other male figures in her life from her crew more rightful to the claim of being a parental figure than Mihawk does.
I think some people take the title ghost princess too seriously. Do not get it wrong, Perona is Moria's Ghost Princess and has been a pirate since she was presumably a child. Of course her epitaph is ghost princess. But if you see the way she is treated by Moria, he lets her run the place. There's signs of her everywhere, she shows up to his meeting like lol boring you woke me up for this? He lets her have a pick of whatever servant she wants and I'm pretty sure she has Thriller Bark’s master bedroom. She's like Regina George meets Usagi meets a pastel goth pirate.
This also brings me to Mihawk calling Perona "musume". I think Mihawk calling her "gosuto musume", just like her epitaph, is taken too literally. In the modern context, this might mean daughter, but we've already established nothing about Mihawk is modern. One Piece itself is supposed to be 1500s if we're going to get super technical.
Not only is Mihawk incredibly proper, but he's also a piece of shit. This is the same guy that called Zoro a little green froggy when they first met just to get a rise out of him. We only see him calling her this once, and this was presumably days after they met (a week or two tops). Immediately after she's called this, Perona yells at Mihawk not to give her orders. Contextually, this could be taken that he's giving her a backhanded compliment for appearing youthful, a lot like calling someone "auntie" might be taken as offense to the wrong person, which would indicate he's trying to get a rise out of her and it worked
When they first meet, it does not go too well. Perona was already incredibly lonely to begin with before Zoro was dropped onto the island, she had taken care of his wounds for days on end and spent at least a week playing hide the swords so she had someone to keep her company. Meanwhile Mihawk is mildly irritated his quiet castle has uninvited guests and Perona's demands of him, his blasé attitude towards Zoro going out and trying to get himself killed has Perona deciding that he's heartless and she doesn't want to be alone with him.
Somewhere within two years, this changed. Not only does Perona go back to the island after seeing to it that Zoro didn't get lost trying to find his own damn ship, but when it's time to go we get a glimpse of what kind of life they've lived without Zoro around. If she couldn't stand him, that would have been the perfect time to go.
Instead, we see that they have high respect for one another enough to take care of each other. To me, this bickering between them about what they did together isn't a conversation between a father and a daughter, it's a lover's quarrel.
This is your typical “old married couple” banter. She’s saying "you never liked my father and you're ungrateful" and he's saying back that she's the ungrateful one because he's done more than enough for her in return to prove he does care.
I think Perona is trying to start a fight here, to protect her own heart so that she can leave on hostile terms and this bastard reels her right back in with just a few words.
When Mihawk says this, there's a pause in the music from the silly arguing background, to this soft romantic beat, much like one might see in a romantic anime or drama when the couple are having a moment of intense feelings for one another.
Mihawk actually doesn't say much at all, but Perona knows his meaning immediately that if he didn't care for her in any way, he wouldn't have said anything at all.
Perona's personal style has changed a lot in the last two years. She no longer wears her hair in pigtails and appears in a younger style. If anything, it looks like her personal style is heavily influenced by Mihawk's in romantic black, whites and reds. She hung up her crown (presumably because she was under the impression Miria was very likely dead and he was the reason she was a proper princess). Her hat in particular is very much the female version of his, featuring white roses instead of a big fluffy feather. Red lining and all.
Hats in One Piece are so important, and I personally love the idea of him gifting her a hat to make her feel better about hanging up her crown (or because she was stealing his- one of my favorite headcanons).
