Yea, good old store and forward. We implemented it in our PoS system. Now, we do non PCI integrations so we arn't in PCI scope, but depending on the processor, it can come with some limitations. Like, you can do store and forward, but only up to X number of transactions. I think for one integration, it's 500-ish store wide (it uses a local gateway that store and forwards to the processors gateway). The other integration we have, its 250, but store and forward on device, per device.
In many places it's also possibly just a left over feature from older times. I worked at a major UK supermarket in the mid-00s, and their checkout system had this feature. But it was like that because that's how it was originally built, it wasn't a 'feature' they added.
Credit card information would be recorded by the POS, synced to a mini-server in the back office (using store-and-forward to handle network issues) and then in a batch process overnight, sent to HQ where the payment was processed.
It wasn't until chip-and-PIN was rolled out that they started supporting "online" (i.e. processed then and there) card transactions, and even then the old method still worked if there was a network issues or power failure (all POSes has their own UPS).
The only real risk at the time was that someone tried to pay with a cancelled credit card - the bank would always honour the payment otherwise. But that was pretty uncommon back then, as you'd have to phone your bank to do it, not just press a button in an app.