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It's probably a legitimate request, to satisfy US Treasury reporting requirements. 

I would advise against providing it by email, though, in case it IS a phishing attempt.  Do it in person, or through the bank's app.

No. Banks will not ask for the information online. I would print the email and take it to the bank.
All my years here I have never been asked. Early on I had to take bank stmts and I redacted my SSN as well as first-year I provided proof of payment of US taxes with redacted SSN

Meeting the request of US IRS is done by filling your tax forms, using your SSN ID, and adding the banking  ID and account number through FATCA forms.

Your Brazilian Bank has no business in this.

What are you trying to do with the bank. If you are trying to do direct deposit of your US  social Security or a pension. Like what was said before normally  the Bank doesn’t ask this only may ask for proof of source of the money. I experience this when wire money for a car purchase. But never ask for the information you posted. Remember Brazil doesn’t have a tax agreement with the US.

I am a new Itau customer. They did not ask for my Social Security Number. The wanted my US passport number, CPF number, and my new CRNM.

Correct, they have no business with the SSN.   They could care less about it.  Only Americans think of SSN as a big deal.

In Brazil, the CPF is asked at all times to validate transactions or provide access to all types of data, , there's hardly any way to link CPF and banking or CC  account numbers  for theft identity and online crimes.

If someone is soliciting the SSN through ITau, it sounds sketchy.  Sounds like your data is getting a read, and you are getting a spoofed e-mail from a scamer. 

Best thing to do, if you do online transactions, is to get a VPN, clean your cookies and browsing history with more frequency, and avoid using Chrome as a browser.   

There isn't to say that illegal data mining is not done here. It's actually rampant.  I used to be solicited lists of people and mobile phone numbers parsed by occupation, income, zip code, auto ownership, etc.    Your data here is fair game, and no one talks about it.