Good! Second homes in any region with low inventory should be taxed...
It seems like a place like Manhattan might benefit from first up-zoning the low slung sections that are currently 4-8 stories. My point here is that I'd start with trying to build enough housing before spending political capital on marginal things that neither unlock supply nor generate much revenue.
The people that have second luxury homes in new york are the people that spent through the roof to avoid mamdani being elected