I hope the info we share on here, on this public forum, will help all others who plan to reach a hotel, garage or other commercial activity in the center of Florence.
The city offers hotels and parking garages the "ease" of computer software to send license plates to the authorities to place their guests and customers in general on the white list, as it is part of them being able to conduct business in the historical center and thus not "be punished" for their location within the ZTL! 
It isn't an automatic exemption, the hotels and private garages have to very consciously ask their customers for license plates if they know you drove into the ZTL to get to them and send the number to the authorities - and, if they don't know, all they need to do is ASK guests as they arrive! 
Your hotel is very close to the SMN train station so it is possible many guests just simply arrive at the hotel by walking there. But when you arrived, and especially when you left, since you say they knew you were going to the pick up your car to pick up luggage and passengers and leave, they should have asked for your license plate. 
It seems to me that the problem occurred here at check out time - and I'm sorry to say that I know many Italians who work in hospitality don't express properly in English which could have led to miscommunication. 
I'm rereading what you write:
	
	
		
		
			The hotel frontdesk staff knew I entered ZTL but asked me to register, I said I am your guest, you are supposed to help me to register, for I experienced the same in Verona 2 days ago, where the hotel staff did all for me simply without my knowing it.
		
		
	 
and I'm thinking the miscommunication happened when they "asked me to register, I said I am your guest, you are supposed to help me to register".
I'm just guessing at what might have been the intention of the staff. It doesn't seem that they asked you for your license plate again, and instead asked for a very generic "registration". I would also not have understood what that meant, since of course, you we're already a guest! So registration was the incorrect word to use, you were definitely not in need of registering anything. I assume you only got one fine for the day you were leaving and not for the day you arrived, so the license plate was correctly sent to the authorities on your first day of arrival. 
Did you drive at all during your 3 days in Florence or was the car parked the entire time? This is an important detail that everyone needs to know about: 
every single time you DRIVE INTO THE ZTL, your car gets photographed and it counts as an entrance that needs to be cleared by someone who puts you on that famous "white list" that the authorities use to take OFF cars that have been logged as having entered the ZTL without a permit. Let's say you enter the ZTL by taking a wrong turn and then exit the ZTL because you can't find the right address, you keep going around in circles and keep re-entering the ZTL as you drive around -- every single time counts as a possible fine, if no one sends your license plate to the authorities to basically say you entered the ZTL to get to them. 
The parking lot where you say you went and picked up the car on day of departure is outside the ZTL so, theoretically, even if you did drive on those days and always parked there, you wouldn't have needed to enter the ZTL to reach that parking lot area.
Hotels don't keep license plate numbers, as far as I know. All they do is enter the number into the system and send it off to the local police so it goes on the "white list" of vehicles authorized to enter the ZTL to get to the hotel. The staff on site at the time of your departure 
should have clearly asked 
for your car's license plate to ensure it was 
sent to the authorities AGAIN. and made it clear that the "registration" was for the car. Here is where I think the error in communication happened, at least taking into consideration the words you have written. 
More things to consider - although 4 months have passed so it it might be hard to remember - do you remember giving them your license plate again at that time, when you were leaving? Do you recall whether they confirmed the license plate had been sent to the authorities? 
I hope you understand that, at this point, it isn't a matter of whose fault it is. I don't think you will get your money back considering the hotel staff is blaming you. You have written about your experience here and you can keep doing reviews on other platforms about the service you received at your hotel here in Florence. There is a lot of competition in the hospitality sector here in Florence, so if the hotel and other places care for their image, they have to care for their guests too and part of that is coming halfway at the very least when problems arise. 
My hope is that your entire experience will not be ruined and you will come to remember your time in Florence with fondness, particularly having seen the sights, art and gorgeous views the city has to offer. I hope you had many more positive encounters with all the people you came into contact during your visit in Italy, so that it can outweigh any of the bad ones.