These retailers state in their own policy that a receipt is not required to make a return. That is less common than shoppers assume, and it almost always comes with a different kind of refund.
No receipt rarely means no proof
What most of these policies actually say is that a paper receipt is not the only acceptable proof. In practice the retailer looks you up another way: the card used for the purchase, an order number, an email confirmation, a loyalty account, or a gift receipt. If the purchase cannot be traced at all, the outcome changes.
Expect store credit, not your money back
The usual arrangement is that a traceable return goes back to the original payment method, while an untraceable one becomes store credit or an exchange at the current selling price. If the item has since been discounted, that current price is what you receive, which can be well below what you paid.
Identification and limits
Retailers that accept receiptless returns often ask for photo identification and record it, specifically to limit how often the same person can do it. Some cap the number of such returns in a given period. Neither is unusual, and both are set out in the policy rather than decided at the counter.
What to check before you buy
If you expect to return without a receipt, keep whatever else ties you to the purchase: the card, the confirmation email, the packaging. Then read the store page for the refund form you can actually expect.