Hello, I have a bank account in Italy, as I

This is perfectly normal.  FOr two years banks have required your NIN/TIN when opening an Italian bank account,  they are just catching up with the new legislation.  

FATCA is made mandatory by the interbank agreement against double taxation and the anti-money laundering law - It is used to identify the person who has foreign exchange relationships with Italian banks - Non-EU citizens are also subject to FATCA - If you do not communicate the requested data , the bank, to comply with the legislation, must:
1> communicate your name and your non-compliance, to the Italian tax office and to the anti-money laundering tax unit, of the financial police

2> immediately close your bank account in Italy.