Assuming we're talking free markets, then the rent should already be set at the maximum price the market will bear. (If not the landlord is leaving money on the table.) Increasing property taxes is to depress the price of real estate. A simple property tax could also discourage new investment, while a land value tax incentivizes owners to maximize the value of their holdings.
So, treat primary residences as primary residences, whether or not they are the primary residence of the property owner. For multi-unit rental properties, the total value of the property is divided among units proportionately to the rent charged for each unit to assess this.
While it's probably not a bad idea, I don't see it moving the needle much.
I'm discussing ways to achieve the goal upthread of insulating people from property tax uncertainty on their primary residence.
I would expect that the larger purpose of that is to allow full-value taxation on all other real property; of which non-primary-residence residential property is a subset (and a fairly small subset, at that.)
Remember that Prop 13 applies to all real property, not just residential property.
It's almost a double whammy for someone who wants to buy a house, but can't afford the current prices or was outbid by a cash buyer going over asking price. Cheer up as we've incorporated the price increase into your rent.