(Spoilers, pls the game before reading)
Euca, thinks Echo and other wishes are ghosts / ghastly like, but I'm pretty sure they're entities that form when someone has really strong wish.
The strength of that wish is proportional to the will of the wisher
Jinny has a very strong will which makes Halley extremely powerful
So powerful that with the help of a mystery person, she can rewind (I think) time.
I'm pretty sure she is only rewinding time to save Jinny, but she needs Euca if Halley plans to succeed
I think the reason why we have no memories is because Halley and the mystery person were rewinding too far back
(I'm speculating btw, so there is a good chance I'm wrong)
Edit: I just thought
What if Halley rewinding time over and over again to save Jinny is her own wish?
*UNDERTALE MUSIC INTENSIFIES*