You bought ON eBay not FROM eBay.
EBay is a venue for thousands of private sellers, most small and part-time, some B&M stores expanding their customer base, and a few big Box stores doing the same.
Unless the seller is an authorized dealer for the company, well, you can ask, but he has no obligation.
You probably got the email from Dyson? That would have been an automatic mailing based on you sending in the warranty card, in all likelihood.
Which is good, because it means Dyson probably will support any warranty claims you may have in future. Good on them! Many manufacturers will not support warranty claims when the item was not bought through an authorized dealer.