I really don’t understand.
Who said the store owner violated "theft"? Who said he did anything wrong?
There is a well-known condition that in order to exchange a product you need a receipt. If you don’t have a receipt, he isn’t obligated to exchange it.
Did I miss something?

He is obligated to replace it because he sold defective merchandise, and this is a mistaken sale. The fact that you don’t have proof that will convince him does not mean he isn’t obligated to replace it. He can of course choose not to believe you, but if you know the truth, then a person may enforce his own rights.

Why does the buyer need to worry that someone else might buy the product?
The product was on the shelf and he had the bad luck to take it. Does he now need to be more concerned than the owner who put the product on the shelf?

Why doesn’t dina de-malkhuta dina play a role here?

Because dina de-malkhuta doesn’t exempt him; it only means he cannot be compelled, because of evidentiary rules.

As far as I remember, "a person may enforce his own rights" applies only in a case where he would be able to prove his claims in a religious court. Am I mistaken?

A lot could be said about this. On the face of it, it is indeed permitted only when he has proof in hand. But distinctions can be made. The Vilna Gaon writes that a Torah scholar may do so even when he has no proof. Presumably that is because there is no concern that he is mistaken. According to this, where there is no concern of error, it is permitted. Beyond that, here it seems he never really bought what is in his hand at all, since this was a mistaken sale. So the seller owes him an item for which payment was made. To save his property from loss, it is obvious that he may act even without proof—how much more so by an argument from the law of the burglar tunneling in. In addition, the seller suffers no loss, since he gets credit from the manufacturer, and the manufacturer is certainly obligated to replace it, and with respect to him there is no need for proof. Also, the proof needed here is proof for a civil court, not for a religious court.
The basic reasoning in all this is that it does not seem possible to speak here of a prohibition that he violated, since he took what was his. So there is certainly no theft here. If so, what prohibition did he violate? Therefore it seems more likely that the entire prohibition against enforcing one’s own rights is only an instruction to a religious court not to validate the act, and not a prohibition on the person himself.
A great deal could be said about this murky topic, but this is not the place.

But when buying a product there is usually a store policy saying that in the event of a defect or malfunction a receipt is required, both in order to verify that the product was actually bought in that store and not in another one, and also to check the purchase date, since the warranty is usually limited to a certain amount of time, like a year.
In addition, consumer protection law also says that you have to come with a receipt. Is there still any basis in Jewish law for exchanging the product without the store owner’s knowledge?

The store policy is irrelevant. It’s like having a policy to steal, or not to compensate for damages, or not to refund overcharging. Bottom line, I didn’t receive what I bought from them.