I’d like to propose a system to make swapping pets easier. Instead of selecting from the inventory, especially when you have many items and the pet are not in the first 9 slots (accessible by number keys), you could right-click on the swap button to open a dropdown menu with all available pets, and then choose the one you want and instead of the text swap presented in the image created should be the name of the pet. This would also help if you are focused on having a strategy when hatching eggs, for example, swapping pigs more easily to boost pets, switching to bees to boost crops, or using chickens when you have many eggs growing.
Additionally, when hatching a new pet, right-clicking on the hatched pet could display its stats (STR, Abilities, etc.) without having to swap it and place it down to check. This would be much more efficient for players focused on collecting rare pets or hatching many eggs at once.
For mobile users, the same actions could be done by holding down instead of right-clicking.