Lilo being strangely obsessed with Elvis is actually period accurate.
The movie seems to be set sometime in the very late '80s to the mid to late '90s. There aren’t any cell phones, except the ones that the government agents have. The TV is a black and white TV which could indicate either the late '80s or the fact that they are extremely poor in the mid to late '90s. None of the vehicles really give anything away as they all seem to be modeled on '50s to early '80s models of cars. The technology that is actually depicted in the film definitely places it at the end of the '80s, as the earliest it could have possibly happened, and probably the mid to late '90s as a more probable timeline.
There were a lot of people obsessed with “The King” at that time, even fairly young (like 5-7 year old) fans. I wouldn’t be surprised to find out that Elvis’s music is generally more popular amongst certain kids and tweens from 1956 until the late '90s.