To what extent is it correct to refer to a credit union as a bank?
To the extent that the differences don't matter or the distinction adds no clarity. A "bank robbery" that hit a credit union is more or less fine, and answering "which credit union do you use" with the name of a bank, though a bit odd, is entirely kosher.
Be aware that, since a credit union is an institution for receiving, lending, exchanging, and safeguarding money, it fits most non-technical definitions for "bank".
Is it correct to use the verb "banking" when referring to credit unions?
Yes. Given the absence of any similarly appropriate verb form of "credit union", it is fair to use the same word to describe the activities one does at both.
Why did Google call Banking a noun?
Well, based on what I see when I search "define banking" at Google.com, I would hazard because their database lists noun forms before verbs, and their first definition is a noun form. You can click the little down arrow to see the complete definition list, which is a neat little thing I wasn't aware google could do.