Is it really a problem that rich people aren't living in all of their homes simultaneously? No. That is their own business, don't worry about what they do, worry about what you do.
What is next, complaining that middle class families often leave one car sitting idly in their driveway?
It really kind of is a problem because of the effects that it has on the housing market. It's completely different from somebody having an idle car in their driveway because nobody else could park on their property. This is a situation where this is an apartment somebody else could rent and live in and decrease sprawl.
And I stand by my car analogy, a car a rich person has is a car that a poor person doesn't have. Artificially high demand for cars ('artificially high' in the sense of people buying more cars than they need) increases their prices. If you want more cheap cars for poor people, then get more cheap cars for poor people. Don't complain about how many cars rich people have.
It's not about taxing the rich. It's about taxing where appropriate to get the market effects you want.
This is the same reason we have such high tax on cigarettes.
Rich people having multiple homes is a distraction that is engineered to tug on irrational anti-rich sentiment. Tax them if you want, but don't mistake them for the actual problem.
There's a lot of apartments in manhattan that are only lived in ~4 weeks a year and aren't rented out otherwise. I wouldn't be opposed to a higher tax rate for those properties. They're effectively jacking up other people's rents and making them have to commute from farther away.