Business don’t need to issue a physical receipt. Business can chose to offer digital/electronic, hand written or printed receipts. A customer can also chose whether they want to receive a receipt, but as you have unfortunately found out, not doing so may impact one in the case of exchanging a product.
Receipts are one form of proof of purchase. Other examples of proof of purchase are outlined on the ACCC website:
However, since you are making an exchange, a business can nominate what proof of purchase they will accept for the exchange. In your case it appears that the business in question choses only to make exchanges when a receipt issued by them is provided.
While they have the right to do such, it isn’t good customer service/relations to not accept other forms of proof of purchase.
This other thread also discusses whether receipts need to be issued:
which may also have some useful information for you.
