Exactly.
Isn’t that exactly part of the plot for Toy Story 5? Jessie, given away by her original owner decades ago and found in a thrift store in Toy Story 2, goes through two subsequent owners (Andy then Bonnie) and is taken to the house of her former owner in part 5. She remembers everything.
Ah, OK, I haven’t seen 5, so I didn’t know that.
And when she was lonely
I was there to comfort her
And I knew that she loved me
Nitpick: Where Jessie actually enters the story in the second film is in the carefully packed “Woody’s Roundup” classic-toy personal collection of toy store owner (and plot villain) Al, of “Al’s Toy Barn”.
I don’t think we ever (?) learn exactly how Al eventually acquired Jessie after her donation years ago by her former owner. Although being in a thrift store at some point(s) sounds like a perfectly reasonable hypothesis.
Follow-up: The overall chronology of the toy life trajectories in the Toy Story films is a bit convoluted. Film canon says that the Woody’s Roundup show, presumably the origin of the toy characters Woody and Jessie and Bullseye, aired 1949-1957. (We see Jessie’s former owner Emily wearing saddle shoes as she ages out of doll play and into teenage culture, so presumably this happens mid- to late 1950s-ish.)
That gives us something like a 35-year gap between Jessie’s donation and her meeting Woody in, hmm, late 1990s when Andy is old enough to go to sleepaway camp in the 1999 TS2? No clue what that part of her story may be.
Presumably Andy’s dad was contemporary with or somewhat younger than Emily, as Andy was canonically born in 1989, so his dad most likely sometime after 1950? I guess we’re supposed to infer that Woody was just tucked away in Andy’s dad’s childhood possessions for the 20+ years between the father’s and the son’s toy play.
Maybe toys are considered to undergo some kind of (reversible?) loss of consciousness in prolonged periods of playlessness, starting to revert to mere inanimate objecthood, as @Chronos hypothesized. (Which would help explain the universal terror of being stuck in an attic and forgotten, despite Woody’s efforts to put a brave face on it for his crew in TS3. The idea of a toy chest in storage as a sort of long-term memory care facility, where the toys try to entertain and companion one another as they slowly slip out of sentience over the years, is indeed kind of terrifying.)
Yeah, we see that also in TS5 in the multiple Buzzes, before they get airdropped to the playground to find new owners. The one who gets found by the little girl out camping is the first to experience that “toy-ensoulment”. Manufactured packaged toys seem to have just enough individual soul/consciousness to embody their commercially supplied backstory, if any. Receiving affection and play from an individual child, along with interacting with other “awoken” toys, is what produces that individual personality “awakening” for a new toy.
Footnote: When never-loved manufactured toys do exhibit some deeper sentience, it seems to be strongly flavored with duplicity and meanness/selfishness. Look at the selfish determination of Stinky Pete the Prospector in TS2 to keep Woody from getting back to Andy so that their “set” can be sold to the Tokyo museum, for example, or the unscrupulousness of Gabby Gabby in TS4 (whom, unlike Jessie, Woody does encounter in an antique shop/secondhand store) when she demands that Woody hand over his voicebox to replace her own defective one.
I think there’s an undercurrent there that anything prevented from achieving its proper purpose will go evil. Including people.
There’s probably some philosopher somewhere who’s hypothesized that.