In most cases when you want to do a price match a lot of businesses require you to show some kind of proof in the form of an advertisement or even a website from a competing business that displays the exact same item for lower price. A lot of times too a company is willing to simply phone in to the other business to verify if the price is accurate and whether or not there is stock available.
One thing I have noticed people doing a lot more is simply getting scanned receipts from other people who bought products at a very good price as proof that the item was sold at the price. Interestingly enough, for a lot of businesses that is good enough proof to then offer that person the same deal. In many ways it surprises me that it would even work as a lot of company policies clearly state it has to be an advertisement for example and that the item must be in stock.
This could simply be a case by a case basis too of some stores simply having managers that are more open to wanting to compete with other businesses as much as possible. Either way, it’s another option to show the business proof of a competing store price in hopes to get you the best deal.