My favorite kind of pet is a cat. ==> I think "my favorite kind of pet" is the subject of that sentence.
My favorite kind of pet is a....(Fiderer, Adele. 1993. Teaching Writing: A Workshop Approach. Scholastic Professional Books: New York. p. 17)
This kind of prank is a crime. (an example or incident of crime)
My favorite kind of pet is adog. (a member or subclass of the dog species)
Grammar doesn't always reflect meaning. It's true that the noun phrase 'My favourite kind of pet' and the noun 'dogs' are the same thing in your mind. It's also true that the noun or noun phrase that comes before the verb is called the subject of the verb. The verb has to agree in number with the subject.
Nouns that come after the verb are called 'the object' except when the verb is a special sort called a 'linking verb'.
The verb 'to be' is a 'linking' one. The subject is the same thing as the noun that comes after the verb. The noun after the verb refers back to the subject.
We could express that idea like this: Dogs=my favourite sort of pet, or My favourite sort of pet=dogs.
What comes after a linking verb is not called 'the object', but 'the subject complement'.
This is very different from most other verbs when the subject is quite separate from the noun that follows the verb.
In 'The man kicked the ball', the man does not = the ball. 'Kick' is an action that affects the ball, so 'ball' is called the object.
Even with the linking verbs, the noun before the verb, the subject, determines whether the verb is singular or plural. 'My kind' is singular noun subject in sentence #1, and the noun dogs is a plural subject in #2.