He has a dog, hasn't/doesn't he?

Either sentence is correct.

Your sentence consists of a statement "he has a dog", followed by a "question tag". The question tag seeks confirmation of the statement and is formed (mostly) using an auxiliary verb and a pronoun. The auxiliary is in negative form. If the verb in the statement is in the present perfect, for example, the tag question uses has or have; if the verb is in a present progressive form, the tag is formed with am, are, is; if the verb is in a tense which does not normally use an auxiliary, like the present simple, the auxiliary is taken from the emphatic do form; and if the sentence has a modal auxiliary (e.g. can/could, may/might, must, will/would, and shall/should), this is echoed in the tag:

He's (=he has) read this book, hasn't he?
He read this book, didn't he?
He's (=he is) reading this book, isn't he?
He reads a lot of books, doesn't he?
He'll (=he will) read this book, won't he?
He should read this book, shouldn't he?
He can read this book, can't he?
He'd (=he would) read this book, wouldn't he?
He'd (=he had) read this book, hadn't he?

HOWEVER:

A special case occurs when the main verb is "to be" in a simple tense. Here the tag question repeats the main verb, not an auxiliary:

This is a book, isn't it?

If the main verb is to have in the sense of possession, either solution is possible:

He has a book, hasn't he?
He has a book, doesn't he?

Thanks, centrox, for the detailed reply.

If I'm not mistaken, "am"(that was mentioned here as a word taking place in tag-questions) doesn't work within this topic at all. We can't say "I'm good at it, am not I?" We should say "I am good at it, aren't I?"

Though "am I?" as a tag exists.