What is the correct term?

A tautology says that if A = B, then B = A. While this is the essence of all mathematics, and all dictionaries, it doesn't hold for rhetorical claims, like yours. If all dogs are animals, it is not logical to say that all animals are dogs. The error you illustrate is a tautology.

The other fallacy that is similar is the syllogism, that says if A -> B and B -> C, then A -> C. So if all dogs are animals and all animals have lungs (thank you, harwarda), then all dogs have lungs. This is not a fallacy, but the form can lead to fallacies. for example, if all dogs are animals, and all cows are animals, then all dogs are cows.

You're talking about a syllogism.

All animals have lungs.

Dogs are animals.

Therefore, all dogs have lungs.

You can look up "logical fallacies" to find the names of the different misusages of syllogisms.