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.

I've had the same, not keen but those are the rules. I've tried to reply to my branch but the bank want me to attend in person so a job next time in Italy.

It's a shame blind trusts owned by British ultra wealthy don't have to do same with their offshore wealth.