TL/DR: 10 Days in New York City is absolutely no problem in any way.
However, what you friend my be referring to is a collection of residency and rental rules that usually hit around the 30 day mark.
This is the distinction between a short term and long term rental which brings different rules and taxes.
Also, New York has some wonky tax and residency rules. I recall (fuzzy) a company I worked with would not allow non-New York residents to stay in New York for more than two weeks at a time and would not allow them to commute to New York for more than six months.