While playing Stray, the cat may be injured. This page of the guide explains whether healing the cat is available and if using this mechanics is important to avoid death.
In Stray, you can get injured in two main ways. The first option is the Zurks attack, which involves the creatures sticking to the cat. They will start to drain the life forces, which is illustrated by the increasing redness of the screen.
First, you need to shake the monsters off your body by mashing the indicated button and then escape them so they won't get stuck to the cat again. Your cat will recover after some time - the red filter will disappear.
The second type of danger involves falling from great heights, after which the pet may have difficulty walking and running. These are entirely staged scenes that cannot be avoided. You don't have to worry of falling during regular exploration because the game simply won't allow it, and it will block every longer jump.
After a fall, you don't need to look for a medicine or a vet . Just wait for some time and the cat will be "automatically healed" and will be able use all the moves again.