The first response sounds off to me, while the second sounds perfectly fine. So, I will begin with the second:
I would agree that you would say "I wish I had a dog" in a situation where you don't have a dog and something happens that prompts you to say this - perhaps you see your neighbours playing with their dog and you wish you could do that too. You regret that you cannot experience the same pleasure of having a dog in the current moment.
So, going back to the first sentence: rather than "I wish I would have a dog", I might say "I wish I had had a dog", which has the meaning of wanting to own a dog but at a specific time in the past. "I wish I had had a dog as a child" make this sense of it being a past regret more obvious. Interestingly enough, this sentence can also be shortened to "I wish I had a dog as a child" and it would have the same meaning.
This brings me to the conclusion that the second sentence by itself can be interpreted in various ways, and that you can change your intended meaning through adding details. Your intended meaning can also be inferred from the context - from example, if you had been talking about childhood regrets, then it is a regret of the past.