The "6-day Rule" comes from Arizona Revised Statutes section 11-1001(10). This section provides the definitions that are applicable to the article of Title 11 dealing with animal control functions. This specific subsection defines an owner, for the purposes of animal control issues and the statute regarding assault with a vicious animal as "any person keeping an animal other than livestock for more than six consecutive days." The purpose of this definition is not to cause any change in ownership of an animal that someone holds for more than six days. If that were so, then any kennel that watched a person's dog on a week-long trip would then own the dog. Rather, the point of this definition is to impose liability on anyone who has an animal for more than six days, as it is reasonable to conclude that the person has some degree of control over the animal.
This is clear by looking at the first sentence of section 11-1001, which indicates that it applies only to that article (and the assault statute). The other statutes in that article deal with issues like dog licensing, rabies vaccinations, dogs at large, and bites. None of these statutes deal with deciding ownership of an animal. Thus, the "6-day Rule" is not much of a rule at all -- unless the dog you've been watching for a week happens to bite someone.