The Senator was looking to frame his suggestion as a good thing...

The Senator was looking to frame his suggestion as a good thing for the entire nation rather than a tiny portion of it and used the term 'peoples bank' to describe it. It's a decent marketing idea Scott - the naming that is. If nothing else, it conveys the absolute truth that a government-owned bank is truly owned by all people in a constituency, not just a tiny subsection. The erroneous terms you supplied to the conversation (because you didn't bother to read the OP), 'credit unions' and 'building societies', might in the spirit of the letter be more accurately described as 'some peoples banks'.