Something that's exchangeable can be traded or replaced by something else, like an exchangeable pair of shoes, which you can take back to the store and swap for a different pair.
When you exchange something, you replace it with something else that is worth the same amount or has the same function, and if it's possible to do this, that thing is exchangeable. When you buy something that's marked "final sale, no returns or exchanges," your purchase is not exchangeable. And your beloved dog isn't exchangeable either — no other dog could ever replace her.