Both are natural, but are used in different situations. "I've got a dog" is considered informal and is sometimes said as a way to emphasize the fact when the fact is doubted (ie. Person 1: "Does anyone even have a dog here?" Person 2: "I've got a dog."). "I have a dog" is considered standard English. This means it can be used in all circumstances whether informal or formal. This also means that it is the most common option spoken among the two.

Both sentences are gramatically correct, but it's more natural to say "I have a dog" when talking about your pet.