Purpose animal
The world introduction says some Men wake with a dog and become convinced they bred it: “This gives Man purpose.” Rusty therefore functions simultaneously as a beloved individual animal, assigned world-object and proof that even a supposedly consequence-free system can generate attachment.
A name under repetition
The repeated adjective changes—Trusty, Red Red, Reliable—give Rusty the exaggerated oral continuity of a story being retold while the speaker is trying not to lose the thread.