That got me a bit worried as the hardcopies now have my SSN and date of birth on them and I sure wouldn't want them lying around (identity theft risk).
Identity theft is a legitimate concern, but the good news is that financial institutions are (in much of the world) just about the most highly regulated and highly secured entities that consumers regularly interact with. In other words, they're required by law to keep your data safe, and they're generally very good at doing so. In a sense, it's in their best interests to do so, regardless of regulation. It's not very profitable to have a customer who's identity has been stolen.
I am assuming that they are most likely just scanning the forms and then shredding the originals.
Likely that's the case. Of course, banks do need to keep records on their customers (in most jurisdictions they're required by law to do so), but it's safe to assume that they're doing that electronically these days. And, again, in most jurisdictions, the safekeeping and destruction of records is tightly controlled, regardless of if they're in paper form or digital.
Should I ask them to please ensure that they properly shred the paperwork once they have the application in digital form in their system or would that come across as me doubting their professional standards?
Go ahead and ask, but keep in mind - the person you're talking to may not be intimately familiar with their document retention and destruction policies - such work is usually relegated to a back-office department, not customer-interface staff. That said, I wouldn't worry so much about coming across as doubting their professional standards, because what harm would that really do? It's not like they're going to turn you away as a customer or anything.
Or do institutions typically keep all those filled-in applications in some cabinet?
As above, generally no - they don't keep the paper records in a cabinet. Typical practice would be to digitize the document and store it in a secure repository, then securely destroy the original.
