In short, no, you can't leave your estate to your pet.
A pet is legally considered to be property, and does not have the legal capacity to own property. You have 3 main options to take care of your pet after you pass:
1. Specifically name who you would like to take care of your pet and list them in your will.
-If you do this, you may also want to transfer them some assets in your will to be used to take care of the pet. The new guardian of your pet won't be required to keep the pet either way, but if you leave some money for the pet's care, it is less likely that they will give away the pet because of the added expense.
2. Transfer ownership of your pet to the person you want to care of the pet while you are still alive.
3. Create a trust for the care of the pet. This is similar to option 1, except, by putting the assets that you are leaving for the pet's care in the trust, you can make it so that the pet's new owner cannot get any of the money in the trust unless they keep the pet.
Which you choose depends on your specific situation. Maybe you don't have the energy to fully take care of your pet anymore, so your best option might be to give your pet to someone now. Maybe you know that your child loves your dog like its her own and would be more than happy to provide a great home for your pup. In that case, transferring by will might be enough. If you have some doubts and want to make sure your pet will be taken care of, then a pet trust can give you some extra piece of mind.
The most important thing to do is to have a plan. You don't want people fighting over who gets the pet. Or, even worse, you don't want a situation where no one is willing to take care of the pet.