The abused wife and the neglected son, Jesse is an amalgamation of Walt's whole family in one person which is why he comes to signify both Walt's remaining shred of humanity and conscious, but his metaphorical soul as well. While there isn't anything sexual or romantic in the show, Jesse very much functions as both a son to Walter and fulfills the position of "wife" by being his partner in crime. There's something libidinial to their relationship that keeps it from being purely platonic.
There's also just choices made by the actors and writers that make their relationship pseudosexual. It's a very complex relationship. Every season has some form of a twisted almost love triangle with the two of them in it. Walt and Jane are paralleled. Jesse is paralleled with Skyler actually far more than Flynn. Tuco's comments about them serve implications about what he thinks of their partnership.
Even the term partnership is held sacred in the breaking bad universe and only given great and high importance to really three relationships between main characters. Firstly, Jimmy (Saul Goodman) and Kim are never legal partners but they both refer to one another as partner several times on separate occasions. They are a married couple.
The second is between Gus and Max whom have been left ambiguously gay with textual evidence supporting this, leaving them more gay coded than anything. The term partner is used very emphatically and emotionally in terms to them, in a scene that was a clear parallel to an earlier confrontation between Walter and Tuco. Max pleads for Gus in the same manner. "He's my partner." Is a one oft repeated.
And then Walt and Jesse the breaking bad universes blueprint for partners. And they're contrasted against a gay coded couple and a literal married couple.
Again there isn't anything romantic or sexual to Walt and Jesse in canon but they are coded as a couple at times, and there are instances where Jesse is treated as Walt's defacto love interest. Walt does cheat on Skyler with Jesse, on an emotional level. Jesse's even paired with Gretchen, both having been his lab assistant.
It's also worth noting that Walt and Jesse each have a shadow self and a persona that they both kill. Gus and Gale, both gay coded characters, represent Walt and Jesse's personas; to make a point, Jesse does so with pain and Walt does so with triumph. Jack and Todd are Walt and Jesse's shadow selves, their dark reflections; whereas Gus and Gale were the representations of what Walt and Jesse could be at their best and most efficient, Jack and Todd are representations of what they could be at the worst and most ruthless. This part didn't have much of a point other than it is interesting that they each have a gay coded foil, that's pretty funny lmao
Walt: You do know what a bond is? A strong force of attraction. Covalent bonds, ionic bonds. The coming together of atoms and molecules to form compounds.
Walt: Chemical bonds are what make matter, matter.
Walt: Bonds are what hold the physical world together. What holds us together.
Walt and Jesse are an ionic bond. Much like the battery, they create together. “There are no individual magnetic poles (or magnetic charges). Electric charges can be separated, but magnetic poles always come in pairs - one north and one south.”
What does a battery need to run? An anode and a cathode are separated by an electrolyte. The cathode is where the current flows out and the anode is the reducing electrode that releases electrons to the external circuit and oxidizes during an electrochemical reaction.
One could argue that Walt is the cathode, the place from which power comes in the story; whereas Jesse is the anode, the one that takes that power and creates the combustive series of events that take place. And, further, still, the element that binds them- copper. Symbolic of the money they are making in the drug industry.
This means that Jesse is literally entrenched with meth in Walt's mind because together they are the two that create the stuff, and that makes Walt feel possessive over Jesse in a controlling and unhealthy way. In Season Five A, Walt comes across as this very different person to Jesse, he's kinder to him now that he has him fully under his thumb, and uses this to get Jesse to dump his girlfriend. Another abuse tactic, by the way, separates him from any outside influences. Part of what drove his passion to kill Mike was because he blamed Mike for turning Jesse against him. His cruelest most personal, most unnecessary murder.
Script directions literally have Walt thinking of Jesse as "his". The show definitely paints them as father and son but they also want their cake and eat it too, by also placing Jesse in as a pseudo-love interest, a damsel in distress that literally is obsessed over by several men on the show, thought to be the most innocent and in need of protection, and the only one to show remorse and horror at the things they've done and witnessed. Jesse is without a doubt masculine by character, but within the narrative, he is given the place of "the chick" (literal TV Trope assigned to him lmfao).
While there is no sexual layer to Walt and Jesse's relationship, Jesse does play Walt's, well, for a lack of a better term, partner. Moreso, even, than his wife. Walt takes off his wedding ring before he takes off the watch Jesse gave him. Walt's last act is to save Jesse's life. Jesse's face is the last that he sees. Per the script, Walt's last thoughts are of Jesse and how well he's taught him and how all is right. He goes out with a smile.
Their relationship is so hard to pin down because it is so multifaceted. They love each other, they hate each other, they're father and son, they have the capacity to be lovers, they can almost function as friends, and they cannot stand each other. They care more about each other more than any other character on the show, I believe it could be argued.
