Any person, other than a person holding a driver's license issued pursuant to section 19-505 and a New York state class A, B, C or E license, neither of which is revoked or suspended, who drives or operates for hire a licensed vehicle in the city except a commuter van, shall be guilty of a violation, and upon conviction in the criminal court, shall be punished by a fine of not less than five hundred dollars nor more than one thousand dollars or imprisonment for a term not exceeding thirty days, or both such fine and imprisonment.
Obviously courts can impose fines - do you honestly think the only thing a court can do is throw someone in jail? Fines are not for torts - penalties are, and there's very strict rules around penalties (they can't be punitive and can only be used to recover costs incurred by failure to perform a contract). You have no idea what you're on about.
The process of fining you is literally a court procedure. The city official doesn't have any authority to fine you - they bring a prosecution against you in court, and provide you with the details by which you can either plead guilty in front of that court (possibly via mail or an online form) and take the sentence that the court gives you (which is most likely to be a fine unless the judge has more information), or plead not guilty and make a defense. Anything else would be a violation of due process.
Again, try looking outside your bubble of the US. This is certainly not the case in countries outside the US.
In Australia, after the local regulator/government started fining UberX drivers for illegally operating a taxi service (rightfully so IMHO), Uber told the drivers not to worry, continue driving and they’ll pay their fines for them.