I agree, but I don't think it is a great concern, as we must rely on the reader's common sense. For example, your first alternative makes it possible that only multiple dogs need licensing. If we did not rely on common sense, then we would also be worrying about who needed the license, the dog or the owner.
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The sentence is fine. It literally means people who own one dog need to be possession of a single license. Possession cannot apply to a dog in the legal sense, and that sense is the only way the term "license" can be interpreted.
It does beg the question, what of people who own two dogs? But nobody asked that question, did they?