Is it legal for an airline to show, at the same exact time, two different prices for the same exact flight?

Unless you actually managed to load two browser tabs in a 1-2-1 succession and still saw a difference between 1 and 2 there’s nothing iffy here. Instead you likely did load 1, browse at 2, reload 1 — and it’s now the same price as 2. If it were still lower that’d be beyond weird. Its just caught up. The price you saw on 1 before reloading is a mirage, it doesn’t exist any more. If you tried to book , at the last screen where you actually need to pay you’d see a message of "the price has went up" or similar.

looking from two different tabs in my browser (one in which I was logged in to the airline system, one in which I was not)

but the airline cannot know this since I am not logged in in one of the tabs

Yeah, I figured a misplaced assumption may have been at the heart of this. That’s not how cookies work.

A browser has only one repository of cookies shared by all tabs. Even if you reset your cookies, as soon as you logged in on the other tab, the airline now knows "that cookie is you".

You got different answers, however, because the two screens were giving data based on a different set of facts.

• I suspect the non-logged-in screen is simply one you left up, and you did not refresh it. Thus, it was more stale than the screen from being logged in. Unbeknownst to you, facts may have changed in that time, such as another customer buying out the last low-cost luggage slot.

• The non-logged-in screen may have presumed application of a new-customer discount. Same way my storage unit site will offer anonymous "me" a month for 99 cents, but after I log in, they want $250! It’s a teaser deal for new customers, who I ain’t. (I actively clear my cookies between sessions).

• It’s also possible the deals are not as equivalent as you think. There are a lot of details to a travel booking, and getting every single one the same for an A/B comparison is hard! So maybe the anonymous screen (if it is that) assumed one tier of luggage option, yet after you logged in it quoted you another tier since it knows you used it the last time.

In my experience, and given the way most booking systems work (especially those of the "traditional" airlines, which must be compatible with travel agents systems, the same price for the same service will always be presented to everyone in the same conditions.

A few things which can affect ticket prices:

• Quotas ("buckets") for lower prices have filled up, so you have to book a higher fare.
• Specific conditions for lower fares are no longer met: some fares are subject to "book X days in advance" conditions, for instance.
• Likewise, if you’re booking a return, if you change the return, that might change the pricing for the first flight as well (because of rules like "stay at least X days at the destination" or "stay the night from Saturday to Sunday"
• Promotions have expired
• You have a different frequent flyer profile, so you have different perks (like free or cheaper additional luggage…)
• The same flight as an outbound or an inbound may be priced differently because pricing is usually done based on the country of origin of the first flight.

For the exact same full combination of flights (i.e. not just a single flight, but both outbound and return, or more in complex situations), in the same class, for the same date, at the same time, I would be surprised if there was actually any difference based on prior history.

Low cost airlines which only allow booking through their own systems may be able to pull off things like that, but I really doubt they do, things are already complex enough as they are.

For additional services, the chances different prices are given are even lower, as those are usually described in detail on the airline’s website. However, again, there may be conditions like "price X up to Y days before departure, price Z after that", there may be different prices for flyers with status, or if the fare type is different, or if you are booking a return v. one-way and a lot more. But those are usually explained in much detail on the airline’s website.

is it legal for an airline to show different prices for the same flight to different customers at the same exact time?

Assuming the two people are offered the same fare class, this depends on the jurisdiction and the reason of the price difference. E.g., This article implies it is legal in the US to "raise prices because of browser cookies (targeted individually)". This other article echoes it. Other countries may have different policies, e.g. in China:

China’s market regulator issued draft rules on Friday to punish illegal pricing activities, including heavy subsidies and the practice by online platforms of charging different prices based on customers’ purchasing behaviour.

I don’t know if that specific law passed, but you get the idea.

Pointers for EU regulations:

• https://europa.eu/youreurope/citizens/consumers/unfair-treatment/unfair-pricing/index_en.htm#shortcut-3
• Zuiderveen Borgesius, Frederik, and Joost Poort. Online price discrimination and EU data privacy law. Journal of consumer policy 40, no. 3 (2017): 347-366.

Related, on the hotel booking side, from https://millionmilesecrets.com/guides/are-airlines-raising-your-ticket-price-based-on-browser-history/:

The Wall Street Journal investigated claims that Orbitz, a popular online travel agency, was showing higher prices for hotel rooms to Mac users compared to those who used a Windows PC.

Orbitz later admitted that it was, in fact, a user trait that they were targeting. Mac users were able to purchase tickets at the same price as Windows users, but Mac users were also being shown more premium rooms, compared to Windows users who were always shown the cheapest price for a room first.

Of course it is legal. I can set whatever rules I like for the products I sell, including charging different prices for the same things sometimes. As long as I am not discriminating on illegal grounds (eg charging higher rent for an apartment to black people, or parents) I can do as I wish.

What’s more, you can’t be sure you’re looking at the exact same ticket even if you’re looking at the exact same flight. There are these things called fare classes. Even if the airline shows you big buckets (economy, economy+, business) they may actually represent a dozen or more fare classes. (To watch fare classes in action check the screenshots in this answer: Flight tickets: buy two weeks before even during holiday seasons? which prices the same flight over a period of time.)

Why would two tabs be shown two different fare classes? One reason is that tab #1 has "soft reserved" the last seat in a cheaper fare class, and so tab #2 is shown a seat from a higher class. Another, since you mention luggage fees and being logged in, is that your logged-in tab gets one free checked bag and your anonymous tab doesn’t. There are various "they are all liars and cheaters" theories as well, but I don’t subscribe to them and it’s well covered in the linked questions. Yield management is complex to do and even more complex to understand from the outside without knowing what drives it.

But to return to your title question, yes, fare classes are legal. Just as a business class seat (on the exact same flight) can cost more than economy, just as a free-luggage-allowance-free-changes economy seat can cost more than a no-refunds-no-checked-bags economy seat, again on the exact same flight, so seats in two different fare classes with no visible (to you) differences can cost different prices. If you use a different tool to do your searches, or do the extra clicks to see the fare details, you may be able to see when this is what is happening.

On a practical level, sometimes if you let your first tab time out the soft reservation it had on that last cheaper seat also times out and your second tab can now buy it.