Is it allowed…

I really don't understand
Who said the store owner committed “robbery”? Who said he was wrong?
There is a famous condition that in order to exchange a product you need a receipt. If you don't have a receipt, he is not obligated to exchange.
Did I miss something?

He must replace it because he sold defective goods and this is a mistake. Just because you don't have evidence to convince him, doesn't mean he doesn't have to replace it. Of course, he can't believe you, but if you know the truth, then a person is in charge of himself.

Why should the buyer worry about someone else buying the product?
The product was on the shelf and he was unlucky enough to take it, should he now worry more than the homeowner who put the product on the shelf?

Why doesn't Dina Demalkota Dina play a role here?

Because the law of the kingdom does not exempt him, but only does not obligate him due to the laws of evidence.

As far as I remember, "Abid is not a judge for his own life" is only in a way that he can prove his claims in court, am I wrong?

This should be expanded upon. Simply put, it is indeed permissible only when he has evidence. But it should be divided. The Grac writes that the seller can do so even when he does not have evidence. And apparently this is because there is no fear that he will make a mistake. According to the law, when there is no fear of a mistake, it is possible. Beyond that, here it seems that he did not buy what he has in his hand at all, since this is a mistake in buying. So the seller owes him an object that was paid for. In order to save his property from a Pishita damage, he is permitted to act even without evidence (Koch comes in the underground). What's more, the seller has no loss, since he is claiming compensation from the manufacturer, and the manufacturer must certainly replace it, and there is no need for evidence for him. What's more, the evidence needed here is evidence for the court, not the court.
The basic explanation for 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 did he do wrong? Therefore, it seems more likely that the entire prohibition against making one's own judgment is nothing more than an obligation on the judge not to legalize the act, and not a prohibition on the doer himself.
Much more needs to be said on this vague matter, and so on.

But when you purchase a product, there is usually a store policy that says that in the event of a defect or malfunction, a receipt is required, both to verify that the product was indeed purchased at this store and not at another store and also to check the date of purchase because most warranties are limited to a certain time, such as a year.
In addition to this, the Consumer Protection Law also says that you must come with a receipt. Is there still a chance that the product can be replaced without the store owner's knowledge?

The store's policy is irrelevant. It's like they have a policy of stealing or not compensating for damage or not returning fraud. I didn't get what I bought from them.