They retrieved those documents and gave them to the intended recipient. “Obtain” suggests something longer term.

So the story says. But what if they didn't? What if it turned out the person receiving the document from the person picking it up also wasn't the intended recipient. How would they know? And if so they'd be holding a document that they shouldn't have in the first place.

You could easily be a paw in an identity theft play like that.

No, "obtain" suggests "obtain".

There's no sense in quibbling over a vague and imprecise summary of the concept. Legally, fraud requires an injured party. In our scenario here, nobody was injured, so no fraud. Whether temporarily possessing a document to hand it to its intended recipient counts as "obtaining" is not really the pertinent question.

I suspect it may fall afoul of a law about making false statements to the government, but that's distinct from fraud.

That's obviously nonsense. Lying on a loan application is fraud, even if you don't get the loan. You're confusing criminal liability with civil liability, which does require damages.