I've seen a lot of posts comparing Bill and Ford (and for good reason, they are very much parallels and foils to each other) but I haven't seen as much exploring how Bill and Stan mirror each other. Where Bill saw Ford as a tool and maybe even sympathized with him because of their shared experiences (outcast by their peers for traits beyond their control, hungry for knowledge and prestige, isolated from friends and family), Bill sees Stan as the embodiment of everything Bill hates about himself. on the website, entering "Stanley" and clicking enter a bunch of times opens up pages from Bill's perspective about how stan defeated him. Bill maintains that Stan didn't actually beat him, that it was Ford's plan, that Stan just got lucky, but we know from the show that that simply isn't true-- it was Stan's plan and it was Stan that defeated him. It's interesting to note that bill is okay with giving Ford the credit for his demise, but Stan? unacceptable. we'll come back to that though.
To Bill, Stan is simultaneously everything he hates about himself and everything he wants to be. When listing all of Stan's faults, Bill calls him a "side character, a resume-inflating, cheap trick loving, past denying overgrown child protected by failure only by a forcefield of denial and shamelessness". who else do we know who ticks all of those boxes? Bill himself is a side character for much of gravity falls in the real world, but in the context of the show this statement shows his fear of not actually being anything special. sure he was powerful in his home dimension, but we see time and time again that there are other beings in other dimensions that are just as, if not more powerful (the axolotl, for instance). Bill takes credit for liberating his dimension when he really didn't, he "honors" his deals through loopholes and turns if phrase, and he shouldn't be throwing stones in a glass house, seeing as he's the one having a temper tantrum. Bill is also deeply disconnected from his past, if him telling us the story of the demise of his home dimension is anything to go by, and only digs himself in a deeper hole through putting on this cheerful, confident, powerful persona. Stan does the same thing, especially as Mr. Mystery. but the thing about Stan is that he grows and changes as the show goes on, while Bill's mindset is perpetual. They both were cut off from their family because of something they did (Stan messing with Ford's project, Bill by destroying his entire dimension), but Stan allows a new family to get close to him and chooses to make sacrifices to make his niece and nephew (and even his employees to some extent) happy. Bill on the other hand, surrounds himself with henchmaniacs, yes-men who just want to party and will follow him as long as he shows them a good time. Every depiction of Bill is a window for him to look through, and with so many in the mystery shack, it's certain that he sees Stanley, the embodiment of everything Bill hates about himself, getting what Bill thinks he could never have. and Bill hates it because it means that if Stan can grow and change and make peace with his past mistakes, it also means that Bill can too. But that would require Bill to actually be vulnerable and endure the pain that confronting your past (many, many) mistakes brings. He would have to acknowledge and accept that his home is gone because of him, that countless lives have been ruined because of him, and that the reason he has never been able to maintain close relationships is solely his fault. But he won't. And as a result, he will always end up alone, a king of ashes.
I think Bill thought of Ford as a way to fill his emptiness because of their similarities, and possibly also because he saw Ford as a form of redemption. Ford was brilliant and good and just like Bill and Bill saw that and may have thought, if he likes me there's no way I'm a monster. in a way, Bill saw Ford as the only one worthy of killing him because if it was Ford it was just a forgone conclusion: Ford hasn't made Bill's mistakes so he is automatically "better" than Bill and Stan, so obviously he could kill Bill. but to have it be Stan means that someone who has messed up in a manner similar to Bill has the capacity to be better. and that shakes Bill to the core.
