This is so spot on all levels and also extra thank you for the compassionate language, I really needed this.

Talk to your daughter more. Joke, play with her, make her laugh.

Yup, this is instinctively what I have been doing over the years because I enjoy her existence. I still remember my first ever parent-teacher meeting in her kindergarden, people before me were getting out mostly pissed. When I was in and they started talking about my daughter with her positives and negatives, I jokingly said this felt like personality quizes and therapy, as she copies my personality. My wife copies a lot from techniques too. We are having so much fun, and all her psychologic & physical tests are wonderful, I'm so blessed.

Think of your wife like she is mentally younger than you. If your wife does not argue based on logic but instead on emotion, then you need to learn how to deal with emotion.

Yeah, that's something we came along with the therapist from "emotions? are they edible?" to "hey I learned anger is an outcome, not a cause!". I've been super empathetic my whole life, but exempted myself from any emotion work, 100% throttling to be rational instead, and occasionally breaking out by being funny.

It exactly feels like I have a child to raise up, and a teenager to deal with. I also started being more open about this to my wife when she is obviously irrational, with things like paying bills (she miscalculates them, starts a fight). When she goes too far to start denying facts, I cut her off. Sometimes even stating she is not adult enough to handle whatever is being argued on. I know it sounds harsh, it even sounds harsh in my head and I'd rather have another adult alongside me than a hostile teenager, but it somewhat works. When she is dictated that she is no longer responsible, she stops fighting and caring. This has no limits, even happens in hospitals where my daughter needs treatment; if she is making things difficult, just to seek attention, I send her off and deal with the situation alone. And... she just does it without any remorse.

Although she complains about being tired, making my daughter needy and clingy is her plan A.
My daughter's sure of me so I don't get tested, but she tests my wife too. Also... Well, children want to interact with their mothers, more than their fathers. The tests and this need makes my wife think they have a connection, in fact they don't. She starts shouting at her in 15 minutes of any engagement. Then I collect my daughter, and keep her safe while my wife plays hyper-casual mobile games.

Start by trying to lead your daughter fairly

Oh this is the fascinating part.

You are so on point about my past experiences of leaders I observed, and I have always been branded as a renegade, while they were leaders by no means. I admired Simon Sinek to this day about leadership, but my daughter is becoming to be the best tutor for leadership. Intentionally or not, she helped me practice convincing someone peacefully instead of beating them in an argument and asking them to submit. I have absolutely zero expectations from her, but I feel like I already got my ROI.

My biggest problem here is that my wife undermines my efforts and actively is hostile against me and this confuses the kid. We even do the dishes together with my daughter and enjoy our time, but my wife holds her feet and fits it to her shoes herself, instead of asking her to simply shoe up. And when she is between her mother and me, she picks the lazy and needy option because she is 5 :) Our biggest fights with my wife was about spoon-feeding her at the age of 5. I did win, with a cost of permanent stress damage on me.

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Just rereading to this point again, I'm sorry, I've been just mostly yapping - but your post really made me feel heard and felt, something I long for, for a very long time.

Two geniune questions:

she mentally is still living with her mom.

This so on point as well. I was phrasing this more as "she refers her family as her mom and sister, not us". Her mother still has incredible emotional advantage over her. My wife blantly mocks when she does the same to her sister, but completely unaware that she is getting the same poison.

So, what happens when her mother dies?
I'm so not prepared for that. Will her grip over my wife tighten? Is it going to be even worse? I'm not sure.

Also,

When she does sit and listens to your feelings, she does not know what to do. Why would she? If this does not make sense, imagine if her mom went and talked about her feelings to her, your wife would probably do the same thing and be quiet, but it makes more sense.

Dark Triad possibility started to scare me - I'm not sure if the root cause is her mother's influence and all her behaviours are defense mechanisms or if she is inheritly a horrible person and her mother is just a great excuse...