Stray has an unusual main character, a cat, and you may want to custom his appearance. This page of the guide answers the question of whether there is a character creation and an option to change the cat's appearance in the game.
The look of the animal controlled in Stray is presented in the picture above. Unfortunately, in the game you won't find the option to choose a different cat or even change the color of the fur. During the game, you can meet animals with a different color, but the appearance of the controlled cat remains unchanged. So you won't find any character creator menu either.
The only change to the pet's appearance is to wear a cat's backpack after meeting the B-12 for the first time in Chapter 3. However, you can't put it on and take it off freely. It is automatically removed by the game in those scenes and locations where the cat and B-12 will split up.
Additional information - after collecting all 27 memories (the most important type of collectible in the game), you will unlock a backpack in a different color (pictured above). This is a purely cosmetic change, as the new version of the backpack lacks any unique properties.