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One to four units should only be owned by people and the owner should have the obligation to live in it or there should be a radius around their property in which they can't own a second one.
Five to eight units should only be owned by well regulated corporations with the fiscal responsibilities this implies. The alternative would be co-ops.
Nine and more should be under a non profit state corporation that charges rent based on trying to break even only (that's how road insurance for people works around here, price is adjusted based on the previous year's cost to the corporation, it's way cheaper than private equivalents elsewhere in the country).
Your plan cuts out normal people from the most likely reason they would own two homes, a place to live and a vacation property.
I think the simpler and easier solution would be to increase property tax rate per property owned
How? Do you own your vacation property in the same city you have your house?
Even if it was just a 30 miles radius, it would be enough to dissuade most people who own two properties in order to profit from it.