This is quite an ancient myth of flight pricing, and I'd love to see it die, but alas it will not. Your cookies and/or browser fingerprint has basically no effect on flight prices.
In addition, the "proof" provided on that subreddit has quite a few problems:
A) That route is priced ridiculously anyway, and they are looking at Delta, who always price ridiculously. Delta doesn't care about price-conscious shoppers, so they have virtually no incentive to try and segment fares based on browsing history, etc.
B) They're looking at a third party flight aggregator. These are the WORST ways of buying flight tickets. They're decent for _finding_ deals, but 9/10 times, the airline's website sells it cheaper, and the airline's website gives you more benefits anyway (as you're the buyer and/or owner, not the online travel agent).
C) Furthermore, these online travel agents cache flight lookups _very_ heavily. My guess is that the search with TOR routes to a different edge server with a different cache. API calls to the airlines are extremely costly.
The best way of getting a cheaper flight has nothing to do with your browser, so don't worry about it. The best ways of saving money are based on _when_ you buy, and of course what airline. Don't buy on a weekend, and try not to fly on one either. Holidays are similar, except it's not a bad idea to actually fly ON a holiday, but not around one.