My theory for Billy's memorie loss is: The westview/ Billy Maximoff memories was mostly because if the sigil (Who i believe was put to protect Billy Maximoff not William Kaplan)
And the memories of Billy Kaplan are lost because of the head injury he got on the car (i mean there was A LOT of blood on his head, it was going down to his shirt). I mean the doctors did said some memory loss was normal.
Of course we have to consider that (different from the comics) Billy Kaplan is technically dead since the car accident so that is a chance of Billy never having the memories of his first 13 years of life back (and if marvel was more willing to do a more Horror story they would make a perfect character for it).
But either way Billy only have memories of Jeff and Rebecca being his parents for the last 3 years and i will take this little piece of the real Billy Kaplan that we sadly will never get to meet
Billy and Tommy were created in the Hex, where they went from babies- to five years old-to ten years old- to be killed in about five-six weeks.
If the sigil was the reason Billy couldn't remember anything pre-accident (wich i believe it was) That means he just got all of his memories back, both as Billy Maximoff and William Kaplan. That includes 13 years of being the son of Rabecca and Jeff Kaplan and the weeks of being the son of Wanda Maximoff.
We all saw him when he woke up after the car crash as Billy Maximoff he was scared. And ok Agatha was mocking him but she was rigth Wanda chose a town full of strangers over him and Tommy (There’s was nothing stopping her from recreating their Sitcom land instead of that farm we saw her in MoM).
Billy thinking about the Kaplans as his real parents os completely the rigth choice. He was the Kaplan's son for 13 years more than he ever was Wanda and Vision son for.
I'm sorry but Billy identifying himself as a Kaplan more than identifying as a Maximoff is the closest we got of his Comic backstory so far.
And honestly? Considering how they are handling him in this show have this little bit of Comics!Billy was a suprise, but a very welcomed one.