It’s because of the verb “wish.” We normally use this verb in the past tense. When we say “I wish”, we are hoping that something different had already happened in the past. You can say “I want to have a dog.” You can also use “hope” if you are talking about the future, but you have to say when you hope for it to happen. “I hope to have a dog someday.” “In 10 years I hope to have a dog.” You can use “wish” in the present tense if you are making a wish. For example, if a genie asks you what your wish is, or if you throw a coin into a fountain to make a wish, you can say: “I wish for a dog."

This use of 'had' is a subjunctive form, used for a wish or a (maybe impossible) desire. I don't have a dog, in reality, but I am imagining a hypothetical or unreal situation in which I had a dog. Don't confuse this with the past tense 'had' - the form is the same, but the use is different. This is not an English past tense. Also, don't confuse this sentence with statements such as 'I want to have a dog' or 'I'd like to have a dog', which are about 'real' situations, and use the infinitive.

Verbs put in the past tense can also express a wish or unreal situation. It's technically "past subjunctive", not "past tense". In fact, quite a number of languages do this - it's just a little harder to spot in English. "I wish to have a dog" looks towards the future, not at an imagined present. You would only say this if the other person were able to do something about it.