they certainly need to report their own residence in order to live there.
This is incorrect. There is no requirement to register with the place of residence in the United States. People who are not citizens of the United States need to comply with immigration law in order for their presence in the United States to be lawful, and there is indeed federal alien registration, but that takes place automatically at the consulate as part of a visa application and at the border when the person's entry into the US is recorded.
But what about tax returns or taxes in general?
Are tax returns based on incomes or are they based upon wealth as well?
The US taxes incomes. Local jurisdictions generally tax real property. New York City is no exception. There will therefore be property taxes on the apartment.
Income taxes cover unearned income, so if the money "in the bank" generates interest income (or capital gains if it isn't literally in the bank but invested in some asset) then that income will be taxed.
A tax resident of the US is liable for taxation on income from sources outside the US, so leaving some of the money invested outside the US will not relieve the person from liability for taxation on income from those assets.
Anyone thinking of doing this should retain a tax lawyer for specific individualized advice.