Gotcha day gifts default to bandanas because bandanas are easy to ship. The person who drove home does not need another bandana. They need the day kept somewhere that is not a camera roll they never open.
What people actually retell
The retold story is short and physical. The carrier was heavier than expected. The shelter name lasted one evening. Someone slept in a shoe. Those are facts a bowl cannot hold. A pet origin comic can, if you write the animal as an animal: size, coat, the ear that folds, and the habit that now decides where everyone sits.
Do not make the pet give a speech unless that is honestly how your house tells the story. Most houses do not.
Who it is for
Give it to the person who was in the car, or to a child who can recite the story and has never seen it drawn. It is a weak gift for someone who wants a new bed for the dog. Buy the bed. This is for the anniversary of the arrival.
What to leave out
Leave out breed trivia you looked up and any magical origin you did not live. If you also read to a child at night and the same animal is in that ritual, the bedtime adventure is a second book, not a replacement for the true arrival.