How can this aspect of product ownership be improved?
The obvious answer is electronic storage (except of course for the packaging). Product instructions are increasingly made available online in a downloadable form (such as pdf), which is a most welcome trend. However, finding the instructions online requires you to know the exact make and model. A possible alternative would be to place a QR code or equivalent on the product, which directs a smart device directly to the instructions. Warranty documentation, where generic to a product, can be similarly made available.
Proof-of-purchase, product key and more specific warranties do not lend themselves to storage online due to specificity to the consumer. A possibility would be to email the customer the information on purchase, as electronic storage (in an email account) and retrieval is far easier. Another possibility would be to store this information in a form of electronic storage (e.g. micro-SD card) that is then inserted into the product. A lower-tech alternative would be to record the information on a plastic card which is inserted into the product.
Another possibility, again based on having the information stored in the product, would be to have the product able to display the information to the consumer, e.g. via an eInk display. This could be used to display a QR code to access the link to further information (e.g. instructions) online.
These techniques would mean that the information is not stored separately from the product, which inconveniences the consumer when they come to use that information.