In class on Monday I am going to do a new exercise where I ask the students to imagine that it is 1908 and the model T has just been introduced. The question is how many new businesses can you think of related to the model T. The point of the exercise is to get them to think about new business ideas related to pricing, sales and distribution. After that exercise I am going to put up the Google logo and ask the same question. For the later question I expect total silence. Google is probably the equivalent of the model T as an economic engine but I doubt my students have thought about it sufficiently well to consider all the possibilities.
Now to answer the question posed in the post title. The obvious answer is "why not". In other words why are dogs full-time possessions. They are not children where you have a moral and legal responsibility for full-time care, love, etc. We cannot time share dogs because pre-ordained constraints have prevented us from thinking about dogs as something where we have interval ownership just like condos in Jamaica. If you relax the constraint that ownership of dogs is full-time, you eventually arrive at time share dogs. Pre-ordained constraints, otherwise known as assumptions, are the biggest hindrance to discovering new business ideas. The other big constraint is pricing models. Everyone appears to be genetically programmed to "sell for cash" at time of sale. The time share dogs are really an example of changing the pricing model.
(Note: you handle the vet bills for the dogs through dog health insurance, which is another great example of a new business idea where all you had to do was relax one constraint--humans.)
For more information on dog time share, read this article from the Wall Street Journal.
My new book, Billion Dollar Company: An Entrepreneur's guide to business models for high growth companies, is available on Amazon. See the fourth strategy Porter should have added, determine if the market opportunity is large enough to interest venture capital and learn the 5-step process to really develop a business model. Book website.