One of the last things Elisa teaches the Asset in sign language is "You and me together". At the end of the film, when Elisa has brought the Asset to the ocean so he can go home, he starts acting distressed and signs "You... and me... together", to which she tearfully replies she can't, resulting in him refusing to leave her. Doubles as a Tearjerker.
Elisa flat-out refusing to allow the Asset to be a test-subject, hatching an elaborate plan in order to free him from confinement.
The fact that she is helped by two people who don't think it's a particularly good idea to steal a creature they don't quite understand from a government facility that will shoot them on the spot for trying, but help her regardless because of Elisa's love for the Asset.
She's also helped by the mild-mannered Dr. Hoffstetler, a Russian spy who had been tasked with killing the Asset in order to prevent the American Army from gathering intelligence but can't stand the idea of hurting the Asset at all.
Giles's opinion of the Asset? Rather than being frightened or disgusted, he proclaims, "He's beautiful!"
Elisa and Giles, her neighbour, regularly re-enact the choreography of famous musicals together while they're sitting on the couch watching said films. Both of them clearly have very romantic personalities but are lonely due to society rejecting them, and so have formed a bulletproof friendship because of it. Giles is even comfortable enough talking about his crush on the man who owns a local pie shop with Elisa.
The fact that the people who help the Asset escape are all members of a minority group. Elisa is a woman and disabled; Zelda is a black woman; Giles is an aging homosexual man; and Dr Hoffstetler a.k.a. Dimitri is an idealistic foreign spy disillusioned by his country's beligerence. All of them are marginalized by their community, all of them suffer from some sort of profound loneliness, but all come together without question to help another who needs it more than they do.
At the very end of the film, the Asset uses one of his abilities to heal Elisa's gunshot wounds while giving her gills where her neck scars were via a Magic Kiss, so that they can have their happy ending together.
Elisa and the Asset's first days together, in which she brings him eggs and treats him respectfully. After being kidnapped from his home waters in South America, brutally chained and caged and electrocuted, his sheer delight at hearing music is wonderful to see.
Elisa and the Asset swimming together in Elisa's flooded bathroom is amazingly sweet, and arguably one of the most romantic moments in the film. Especially when it ends on the two of them holding each other tenderly.
Dr. Hoffstetler beginning to form an Odd Friendship with Elisa and Zelda over their mutual concern for the Asset, to the point of dropping his accent and telling them his real name. It makes his death at the hands of Strickland that much sadder.
Elisa signing to Dr. Hoffstetler "You're a good man" after he helps her escape the facility with the Asset, and gives her a saline mixture to help keep him alive at her house. Likewise, Dr. Hoffstetler tells her it's been an honor to meet her. It's not uncommon for the entire theatre to erupted in "awww" at that scene.
In a way, Giles not being upset that the Asset ate one of his cats.