That’s an urban legend. Prices are aggregated and cataloged by ITA matrix (purchased by google/alphabet a decade ago). The only time a price change is when the airline itself changes the current retail price.

Now, if you’re using a third party service - like Priceline or Travelocity - to check and book tickets (which is a terrible idea, btw) they may track your history and alter their price, but the master index (served by google flights from ITA matrix) will not change for you, personally, or your specific ip or other identifier.

Note: Airlines will adjust their prices based on interest/purchases, so if you’re hitting a flight with requests to bill so hard that the airline thinks the flight is popular, or you go through the reservation process far enough to “fill” a particular fare class, then the price will change - but it will change for everyone, not just you. Similarly, Amazon will raise pricing during a buying surge - but it’s for everyone, not just you.

On the (mostly) plus side for untracked browsing - as long as it’s so tight that you’re hopping ips and avoiding any back end fingerprinting of your system - Many merchants do source-based pricing. Ex: if you go to book certain services direct vs following a referral link (like via a cash back site or association) you my find different pricing. Using and AARP link to some travel services will result in a 25-30% price increase, to offset the 20% rate coupon they offer, plus a little coin for themselves. Other sites will also trigger cost basis alterations - especially for services which are hard to identify or compare a fixed cost.