Not as explained
For whom a serial number is proof of purchase depends on their access and data handling.
Let me give you an example:
I just sent in a computer for repair. It has a serial that is unique combined with the manufacturer's name. This serial was also marked on the initial bill of sale by a third-party vendor, denoting it as an individual item. The seller has a database of sales (combined with the customer number and identification of model) and could locate a copy of that bill of sale matching the serial number and then could provide a facsimile for warranty purposes. The manufacturer required the date of sale for warranty purposes, even though they could estimate the sale to have happened between the making and the registration.
To that specific vendor, the serial number, combined with proof of who you are, would be enough proof to show that you purchased the product. However, not all stores store specific serial numbers for all products, if they are available at all. For example, the same store also sells flash drives without serial numbers, and they can not be identified but for their type description.
On the other hand, the registration would not help on its own: while the registration allows to narrow down the date of sale and might help to identify the bill at the vendor, it doesn't prove anything that is not on the registration form. If the registration form does not require showing a receipt, it can not prove a sale but only that it was registered on a certain date by a specific person or group.
The manufacturer might identify their wholesaler via the serial number, who in turn might identify the individual store that got the item, and that store might then identify the sale if they have sufficient information. But on its own, the registration at best narrows down the time of sale, unless some paperwork of the sale was attached to the registration.