This is state by state, city by city. In some places, a mayor has broad powers over agencies, taxes, etc. That isn't quite the case in NYC. It's up to the state to delegate these powers, just as the federal government delegates powers to the states.

NYC is explicitly restricted (relative to other cities in NY) by the state in terms of what it can do. It can't independently pass its own tax laws (in many cases, at least), which other cities can, for example. Multiple agencies that would often be municipal are handled by the state or require state approval/ explicit delegation.

The city also gets exceptions for more power, including taxation powers. It's all case by case.

The NYC mayor's powers are complex for this reason. On the one hand, no one cares much about other mayors, so you have a ton of political power. On the other hand, you're not exactly empowered to do a lot without asking someone else to sign off.

While the federal government does delegate some powers to the states, many of the states powers are reserved to the states explicitly in the constitution, with the federal government only having those powers explicitly granted to it. (See the 10th Amendment where this is explicitly laid out.)