Estella was her Twin Sister AI that she never meet prior to being discarded.
At first, she despised her sister for being chosen as a better AI, and being discarded when the experiment failed. After Toak found her still moving on the AI junkyard, they gave her a new mission which fueled her hatred for her sister even more. After reconciling with her sister, they both sacrificed themselves to avert the impact of the sunrise hotel and break it apart to minimize the casualties.
Many years later, after she was given a new body, Elizabeth still ponders what was her sister's last moments, as her data memories are from before the sunrise incident. When Vivy recognized Elizabeth, she told that Estella's last moments was being with her sister and smiling.
Vivy and Elizabeth was hostile at first when they first met at the sunrise hotel.
While Vivy was good at combat, Elizabeth has better combat abilities as her, though she was still overpowered when Matsumoto overrides her programming. She still ponders why Vivy was doing other work more than her main mission was and thinks she's a defective AI like she was, but Vivy assures her that her main mission was still making people smile with her singing. Vivy also convinced her to reconcile with her sister Estelle, as she was more worried about her than she knows.
Many years later, they met again, even though her data memories are from before the sunrise incident, she still knows who Vivy was by watching the video archives about the incident. She even asks what her sister and her original body's last moments was, and Vivy assures her that it was with a smile. She even sacrificed herself just to help Vivy on the second timeline.
Elizabeth was very loyal to Yugo, as he was the one who saved her from the junkyard after being discarded, and gave her a new mission to live.
She was very devoted, to the point that she injected her master with a sleeping drug just to stop him from confronting Vivy herself and putting him to an escape pod just to save him from the Sunrise Hotel's destruction.
Though Elizabeth's data memories has been archived by Toak (even if it wasn't implied, it was assumed that it was Hugo who archived her data memory), uploaded to a new, non-autonomous body, and became Yui Kakitani's (Hugo's Granddaughter) life keeper AI, she was still loyal to Yugo as her archived data memories was in before the sunrise incident, though she got a new mission when Yui played Hugo's last will to Elizabeth - to save what's important to him, though he didn't implied the reason, she still follows it until the end.
Elizabeth served Yui as her Life Keeper AI, though Yui also thought of her as a friend.
Though not much has been revealed about Yui's past, it was implied that she was Hugo's Granddaughter, and she gave Elizabeth a new body after finding her data memory in Toak's Archives and becoming Toak's Moderate faction leader. As Hugo's last will was to save what's important to him, she protected her with all her life, and pressed on even though she was devastated when she was killed on the second timeline.