When you do a search on a travel website (including most airlines official sites) you are actually being served results from a GDS. These companies (Worldspan, Sabre, etc) pull in airline/hotel/etc availability and produce an a search and fulfillment API. They are the reason you can get a flight that connects across multiple different carriers.
I've built a number of successful OTAs (Online Travel Agencies, aka websites) and never once been asked to provide visitor IP addresses or cookies.
In the early days of the internet doing repeated searches would increase prices because these systems were designed for travel agents to do a small number of searches, and a spike in searches was a demand signal. Almost all demand based pricing has been eliminated from air and hotel because of internet "casual shoppers" and price wars.
Do a google search for "clear your cookies before booking flights" and you can read all about it.