HSBC threatened to close my business account, after 5 years that had started with a face to face meeting and proof of identity. They put a nice big notice on the website so that the primary user could see when he logged in. Unfortunately, all the day to day stuff was done by the secondary user, for whom there was nothing, not even a "tell the primary user to log in" notice.
So they wanted the usual, passport, driving licence etc. and it got sorted. The real people were much nicer than the "tell us who to send the balance cheque to when we close the account next week" letter.
Squareup had a different version of paranoia. We signed up, and got it all working, but didn't have an actual reader as all we wanted was the over-the-phone payment facility. I put £10 through, and checked that the £9.75 arrived in a day or so, which it did. The next payment was around £1000, which went through fine, but had obviously triggered the "possible drug deal" alarm, so Squareup kept the money until driving licence, passport etc had been scanned and sent.
It wasn't too bad, in that the customer never noticed, and we didn't have to do it again. Squareup online is generally quite good, as the card numbers are entered directly onto a really simple page, with no need to write anything down, and they will email the customer an acknowledgement if you want. It means that we never store customer card details, which avoids the need to secure storage and monitor access and all that worry. The rates aren't bad and there is no monthly charge.
At least banks will accept scans of driving licence etc. Some time ago, one organisation wouldn't accept a paper copy of my licence because it was twice full size and in colour. They wanted a photocopy, which had to be in black and white, and it had to be the right size, in spite of the fact that real photocopiers were already a dying breed, being replaced by printers / scanners in one box.