Is my pet worth anything?

Pets currently have no purpose (except making you go d'awwwww when you get it). It's been request by the fanbase that they be given a purpose in a later update, and the developer is considering adding new ways to interact with them (though note that "interaction" doesn't necessarily mean "purpose").

Rather than a purpose though, it does have one minor effect:

Its happiness is part of Grandpa's scoring calculation.

Apart from those, currently the pet has mostly vanishingly subtle, subjective, or aesthetic purposes:

• It gives a new farm a sense of activity and life, which was the original purpose:

  My original purpose for the pets was to add some "life" to the farm in the early days before you have chickens, etc. They are supposed to just be cute aesthetic creatures that don't require any work. But I will try and think of ways to make them more interesting without straying too far from the original purpose.

• It gives you an early lesson in how the game's animals work, before you have ones you need to take care of (i.e., that they block your way, they can be pushed if there's room, they're sort of random and not very fast, short ones are easily visually lost in grass, they won't damage your crops, they can be interacted with, etc.)

• Players who have a penchant for roleplaying get obvious benefits from its presence
• The choice of pet expresses the player in screenshots

There's currently no way to get rid of your pet, and I get the sense that the developer has no intention of making it possible.