This is honestly just my own additions to a theory video I watched to make it stronger:
Note: There’s been claims that the Re:Zero short story of Emilia’s perspective debunks this, but that event happens between Arc 2 and 3 and not Arc 1 according to the Re:Zero wiki.
What if it is possible that the person that Subaru met the first time that he went into the Isekai world really was Satella and not Emilia? Now, I more follow the anime than the Light Novel, but I’ve looked into spoilers and it’s only made me think this theory might be plausible, so please here me out.
After re-watching Episode 0, Puck never says the half-elf girl’s name and when he mentions that she has terrible taste, this is after the silver-haired elf looks over Subaru’s hands and concludes that he is part of a high-born family because his hands aren’t worn. It’s ambiguous enough that Puck may have been referring to the silver-haired half-elf’s taste in Subaru. Some might argue Puck simply being there with who we believe is Emilia sufficiently ruins this theory, but Puck’s limiter means his old memories were sealed away so he’s not reliable in determining whether Subaru’s first meeting was really with Emilia as we all presume. Subaru repeatedly calls her half-elf appearance cute, says she’s a very nice and thoughtful person, and that he insists on keeping his promise of finding what she’s lost. This would be consistent with Satella claiming Subaru is the only one who gave light to her life and a purpose to live. It’s worth noting that when “Emilia” states that her name is Satella, Subaru has already been spending a lengthy amount of time helping her find Felt by talking to townspeople and they’re getting along really well. There’s no reason for “Emilia” to lie and say her name is their world’s equivalent of Adolf Hitler as she isn’t showing any signs of seeing Subaru as a nuisance or hindrance to her investigation. Later on, they help the green-haired girl find her mother and there’s even commentary about how they’re like a married couple. Even more interesting, when we see Subaru bleeding out and slowly dying as he watches “Emilia” get knocked down and bloodied, we don’t technically see “Emilia” die and Subaru’s last wish before blacking out is hoping he can do everything that he can to save her, just as he does for Satella later in the series. Flash forward to episode 13, where Subaru embarrasses himself during the Lugnica Royal Selection, Subaru actually brings up the “first meeting” with Emilia for which Emilia obviously has no memory of as his reason for being fixated on her. Emilia mentions that the “Emilia” inside Subaru who can forgive him for anything sounds amazing as a way of pointing out that he likes the idea of Emilia instead of seeing her as an actual person.
While there are some doubts about what it meant in the Light Novel, Trial 2 brought up the possibility that Subaru is drifting into parallel worlds each time his Return by Death ability kicks in. One of those worlds shows Subaru’s body rising with the Jealous Witch taking over. Putting Emilia’s commentary, Subaru’s repeated references to the first meeting, the parallel worlds shown, and the Jealous Witch taking Subaru’s body… could there be a connection? If this is the case, what if his first time in the Isekai world was a parallel world where he really did have a mostly fun day with Satella and he was helping her out? Thus, the very beginning of the story would be the “mystery” of how and where Subaru met Satella. This would also mean his feelings and relationship with Emilia is built on a total lie since he assumed that she faked her name, the first time they presumably met and he had “fallen more in love” when reflecting on her being crazy enough to call herself Satella to him. Consequently, Subaru’s first real meeting with Emilia was by insulting her by confusing her with Satella since he had no idea of the major differences in the parallel world that he had just died in. His final promise of doing everything possible to save her when he died in Episode 0 may have been the trigger for Return by Death and fits with Subaru saying that he wants to save Satella instead of killing her. If parallel worlds are what Subaru’s been hopping across then Satella exists and has access to all of them, so Subaru having a happy day with her once could have the domino effect of her becoming obsessed with him in all possible worlds and latching onto him by giving him the ability of Return by Death to kill her.
Some of this is a bit of a stretch, I suppose. Like why Puck’s memories would even matter, if Subaru is hopping parallel worlds, but since parallel worlds was brought in as a possibility until otherwise conclusively shown to be false, I think it’s fair to make theories from such possibilities. Perhaps one of the final twists in Re:Zero will be that Subaru’s fixation and “love” for Emilia is a complete lie after all and didn’t simply happen due to a reset the first time, but rather that he confused Emilia for the real person that he fell in love with, Satella.