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 having fun 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.
RatherSo, 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 pastin the past. Perhaps "I wish I had had a dog as a child growing up"child" make this sense of it being a past regret more obvious. Interestingly enough, this sentencessentence can also be shortened to "I wish I had a dog growing up"as a child" and it would have the same meaning. Which makes
This brings me thinkto the conclusion that the second sentence by itself can be ambiguously interpreted in various ways, and that you can change your intended meaning through adding detail or context to it will clarifydetails. Your intended meaning can also be inferred from the sense in whichcontext - from example, if you had been talking about childhood regrets, then it is meant to be understooda regret of the past.