Correct word for a rule that force the usage of something to a limited subset

I am looking for a word that describes the following type of rule:

Let's say a customer has received a gift card. This gift card has a rule that prevents the customer for shopping anything in the shop, but only specific categories of merchandise. Let's say only books. So the customer may only purchase books and not the other merchandise like CD's, posters, aso.

The shop would then add a "XXX rule" on all gift cards that should be limited for payment on specific categories of merchandise.

Example usage: Customer: "Why am I not allowed to purchase this CD with this gift card?" Cashier: "I am sorry, but there is a (XXX rule) that specifies only books to be bought with this gift card."

I've thought about restriction, but that is, to my knowledge, generally used to describe what to not allow, and the word I am looking for is more to the essence of describing what to only allow

I am not a native english speaker, so my question may contain strange assumptions due to my limited english grammar knowledge.

"I am sorry, but there is a (XXX rule) that specifies only books to be bought with this gift card."

The correct word is limitation.

Limitation
noun

A limiting rule or circumstance; a restriction.
‘severe limitations on water use’

Restriction would be similarly correct.

I've thought about restriction, but that is, to my knowledge, generally used to describe what to not allow, and the word I am looking for is more to the essence of describing what to only allow.

I agree with you here. Restriction implies a prohibition ("you cannot buy a TV with this coupon"), whereas a limitation implies a selected applicability ("you can only buy books with this coupon").

But this is a subtle difference. Restriction would still be correct to use, but I would indeed favor using limitation for your specific scenario.

Condition (as mentioned by Edwin Ashworth in a comment) is the best option in my opinion.

Condition
noun

A situation that must exist before something else is possible or permitted.

‘for a member to borrow money, three conditions have to be met’

While Restriction and Limitation are valid in this circumstance, it is true that they are more about "you can't do something" rather than "you CAN do something". Condition is more broad and can apply to essentially anything that can be expressed in the form "In order to do X, you must do Y"; in your case, "In order to use the gift card, you must purchase books". It also works to insert it directly into your provided sentence

I am sorry, but there is a condition that specifies only books to be bought with this gift card.

Also, while it wouldn't be used in this circumstance, there is a related concept called a Whitelist, which very distinctly conveys the idea that "Only these things are allowed and nothing else". It is rarely used in conversation though, and has some additional connotation that would make it not apply to this situation anyway.