The case of a man and a dog is different. The man has the right to marry. The dog does not. The man unquestionably must find an equal partner for his marriage who has the right to marry him just as he has the right to marry her (or, according to some, him).
I see a technical difference, but I don’t see why that should make a difference. The man has a right to marry, and wishes to marry his dog.
If you are opposed because you don’t want to force the dog into marriage, we can dicker about that. But really? Is the objection really that you don’t want to force the dog into marriage against its dog will? I don’t believe you if you say that it your only or even principal objection.