... not really. Most folks only own one house, so nothing complicated. A few folks own multiple houses. These folks already need to establish residency in one house for tax purposes in a lot of places: Taxes vary from city to city and state to state. It isn't a new grey area, and we can use this established system to tax. The folks can, in general, just use one house or apartment or pay more for more than one house.
I'd personally only give it out to folks with longer terms: The place should still be someone's primary residence, and only give a discount if it has been rented out x months out of the year. This would be to discourage airBNB units (for example) and encourage landlords to make their units affordable and attractive to the residents of the area.
Alas, that would be a terribly idea, because all of a sudden you have the government snooping into your living arrangements. Especially for grey areas where people share their time between multiple apartments. Way too complicated.
And, people living in bigger places would get effectively get a discount this way.
Your argument does highlight a problem with the transition phase of the introduction of the tax, though. There's a few ways to solve this, eg by slowly phasing the tax in over a decade or more.
(It's not a problem in the long run, because land prices will drop until the financial burden of tax plus mortgage payment is the same as today.)