I enjoy this ship for a lot of reasons, ranging from my personal headcanons to their themes just meshing and complimenting each other's very well.

Truth be told it all started for me with how it left a lasting impression on Wriothesley that Furina didn't stick around for his trial fully. People can interpret and make up why that was the case all on their own. He thinks it's because he simply bored her because he kept pleading guilty for everything but argue that not to be the case since Furina is a character where there's more than meets the eye and little does he know that he also impacted her in that instance. This and it's very similar to how he thinks the chief justice would never show favor for anybody and wouldn't go out of his way to look out for a teenager growing up in the fortress of Meropide. I'm of the belief that Wriothesley's trial (any trial truly) hit Furina hard because he was just a kid who was pushed to the extremes that he was to enact a justice he wasn't getting anywhere else. The trial was straightforward, and it wasn't the trial she was waiting on but also seeing a little kid get sentenced to the fortress the way he is without taking into account the circumstances of why he did what he did was heavy for her, so she excused herself. With that in mind, Furina kept up a fictional character for 500 years, though her acting was good, when her act gets challenged Furina does one of three things most of the time: deflects, avoids eye contact, or she just straight up excuses herself from the scene before her act completely crumbles. With that in mind that's why I feel like she left his trial, because it was affecting her too deeply and it was thus challenging the act she had to put up.

Seeing a kid who you know wouldn't have been pushed to such lengths if there was a system in place to properly take care of him and look out for him would hurt and affect anybody because they're just a kid. Furina is no exception to that rule and since we know how she really is and how deeply she's affected, that is also the case for her in my humble opinion.

Wriothesley sends Furina his finest teas to "make up" for disappointing her. This is information Furina doesn't know, as far as she knows he's only sending her gifts, she has no idea there's a deeper reason behind why he sends her the teas that he does which lines up well with how he thinks she left his trial because he bored her when that is not at all the case. I personally believe that he still sends her teas to this day. I don't believe that stopped only because she stepped down and has retired from her position as the archon. That is just me though.

Post-archon quest, I feel like Furina would find comfort in Wriothesley as he was someone that was not personally in her trial. Yes, the trial was the whole point why she put herself what she put herself through, but we're not going to act like that did not affect her either, let's not be ignorant. Furina's well of love runs and goes to deep, she has no hard feelings on anybody, but I do believe that Wriothesley not being in that trial makes it easier for her to get along with him. There's a comfort in that small yet important detail, it's a little bit of an escape for her. Both her and Wriothesley have gone through a process of rebirth of sorts, Furina more recently than Wriothesley. There's a past that is lost to them now and they have to push forward with who they are now. That's one of the themes that ties them close together for me. Furina has since stopped being Focalors, today she needs to start living as FURINA, the same way Wriothesley made the choice of leaving behind who he was and start living his life as WRIOTHESLEY. The circumstances are obviously different, but it is still the theme of rebirth for the two of them.

I feel like Furina would feel guilt for that trial many years ago, a guilt she shouldn't feel because he doesn't hold either Neuvillette or her guilty for the actions that he took. Furina mentioned to Neuvillette in her story quest that there's so much she wouldn't be able to see sitting in her highchair up above, that she needs to be a member of the crowd in order to understand her people better and just see what they live through and how they experience their day to day. Furina wants to understand what it's like to be a human from their perspective and something that I think would help with that is her seeing the underworld of Fontaine just as much as she sees of the overworld, I think that's something that would align with that interest of hers of wanting to understand them because these are exiles. There's people in the fortress who no longer want to come back up because integrating back into that society is too difficult for them, that would be an interesting subject point for these two, more so Furina than Wriothesley.

These are just some of my rambled thoughts on these two, but it gets the main idea as to why they appeal to me how they do. Then there's the blue and red, the fact that he's mainly from land and her roots come from the sea/waters, she's hydro and he's cryo. Furina is associated heavily with the Spring so there's the point of view of seeing her as Persephone going into the "underworld" (Fortress). Wriothesley and his ties with Cerberus who would dutifully and loyally look out for Persephone in the underworld. Just little nuggets like that that also weave in very well with my vision for them. Thank you for asking!

Anybody can perceive him as gay if they want him to be.

You're speaking of the coding of his character. Coding does not equal implication. Because this Vautrin from this other book is not the Wriothesley from Genshin Impact. Many such cases of course where other pieces of fiction and life as we know it tend to impact other forms of fiction and how we consume it has the tendency of being influenced by our bias.

Let's just touch on a basic. Coding and implications.
Implication = "Man, I really love fries!" Character B said. (B saying they love fries so generally IMPLIES that they must also love curly fries)

Coding = Character B's shirt has an overwhelming number of curly fries as a design on it. (B wearing said shirt makes you THINK that they must love curly fries)

This is the difference between the two.

Wriothesley also has staple-like attire similar to that from webtoons that tend to include a male and female lead (romantically most of the time) which is something I have seen multiple people associate him with along with what you have just told me.

He can be gay if you want him to be. I think he's queer because that's what he makes me think.

Am I right? No. Am I wrong? Also, no.
Sexuality isn't a personality trait though. He is or he isn't, he's still a character that appealed to me and I enjoy very much regardless.

Directly correlated no, as Vautrin and Wriothesley do not share any sort of deep relationship and they don't depend on each other for there to be substance for each respectively. If you're asking me if there's parallels between them as characters absolutely there's a few within the canon setting of the game.

• Vautrin enacted his own justice against those who forced Carole into her unfortunate fate. Wriothesley enacted his own justice against his foster parents who treated him and his siblings like cattle.
• Both committed their crimes in the name of those they championed as family. Vautrin saw his little sister in Carole (Vautrin's sister had since passed away). Wriothesley had his siblings.
• Both were failed by the justice system in place in Fontaine, both in their respective times of date. Since Vautrin has long since passed given the timeline.
• Vautrin and Wriothesley share a special relationship and bond with the Melusine. There is a respect both have for the Melusine. An attachment.
• One can surmise that both Vautrin and Wriothesley may share an identical form of thought. Traces of this can be found in the Mutual Aid Network which was formed by Vautrin and in the way Wriothesley tries to run things in the Fortress of Meropide.

My view on those two is the same one as what Vennessa is to Jean and Nabu Malikata is to Nilou, a rough comparison since Vennessa and Nabu Malikata are quite lore prominent but you know what I'm trying to say right? They are but mere echoes of the past in a new samsara since we're all too familiar at this point by how important "time" is in this game. I hope this answered your question, I appreciate you asking.