Can you earn miles with different airlines for the same one flight, if they are part of the same loyalty program?

No,

in the specific case of AAdvantage Terms and Conditions as jpatokal has already noted, but also, no in general. Accrual of points in multiple programs is expressly prohibited by most programs in the T&C, not just airlines but for frequent dining programs (you can have multiple credit cards for one program, but a max of one program per credit card) and rental car and hotel partnerships (sometimes you can get both car/hotel points and airline points, but only one one partner airline). The very economics of loyalty points runs against double-dipping.

The obvious and original purpose of frequent flyer miles was to encourage people to choose one particular airline over its competitors. So, if you could double-dip— or triple-dip, or 30-dip— you’d no longer have an incentive to fly a particular airline. This is the most fundamental reason why airlines have no interest in allowing accrual to multiple programs.

But today, frequent flyer miles are also a huge profit center for airlines— one of which they must be highly protective.

Miles are not "free" even though they seem like a sort of fiat currency that airlines could "print" at will. The airline has no direct interest in where you do your banking or grocery shopping; none of that money makes it into their coffers. But your bank and your supermarket care, and because a significant number of their customers like miles, they will purchase miles from the airlines to give to their customers as an incentive. During the difficult early 2000s, FFPs were often the only profitable part of an airline’s business, and in a large program contribute hundreds of millions of dollars to the bottom line: even money-bleeding Qantas reported a $286 million profit in its frequent flyer program last year.

This is why airlines are so deeply in bed with their credit card partners— Delta with American Express, United with Chase, and so on. When Air Canada’s Aeroplan was spun off into a separate company, it reported that 63% of revenue came from financial services, with only 27% from Air Canada (mostly flights)!

Second, your program has a tremendous amount of information about you that is valuable to marketers— where you travel to, what kind of tickets you buy at what prices and when you book them, what seasons you will be in different cities, and so on. This has boosted the valuations of FFPs far above what would be expected based on purely revenue-based pricing.

When you consider this, there is no scenario in which double-dipping of miles would be profitable to any party except the passenger— and who has already given the airlines money in exchange for miles in a single program. Your fare has gone to Alaska Airlines, and you requested BA miles for them. Alaska purchases miles from BA and awards them to you. To request AA miles, Alaska would need to purchase and track miles from AA as well. That might make you and a lot of other people happy, but it would annoy BA and AA, who would be jealous of each over over your business, and it would annoy AS, which would incur additional costs because the benefit would probably not increase revenue enough to offset the administration costs and the annoyance of your partners.

This is also why it is so difficult to change credit after the fact. If you decided you wanted AA credit instead of BA, you’d need to ask BA to cancel their credit, and recognize those miles as still available to AS. Then you would ask AS to credit the flight from their stash of AA miles instead. You can see that such changes would incur accounting and other processing costs, and because credits and transfers are generally done in batches/sweeps instead of real time, there is a risk that customers would redeem miles while they were in the midst of being moved from one account to another.

Double-dipping scenarios

Now, there are a handful of scenarios where you can get away with double-dipping in airline programs, though it’s usually something which seems like double-dipping, but isn’t actually.

Rebooking on a non-partner airline

For example, suppose you have purchased a ticket to fly on Delta, but due to mechanical problems, they are unable to fly you to your connection, and arrange for you to fly on American instead. You give your American AAdvantage number when boarding the AA flight, and receive AA credit for it, but you can also write to Delta and request original routing credit, which they will often give as a customer service gesture.

This won’t work with airlines who are partners, because they will see the credit in each other’s systems and deny the double-dip. If your Asiana flight from San Francisco to Seoul is overbooked, and they put you on the next flight out on United, you won’t be able to request both Asiana and United credit for it.

Non-mileage airline programs

Singapore Airlines has two programs: KrisFlyer and the PPS Club. KrisFlyer is a traditional frequent flyer program, whereas PPS is a Singapore Airlines-only recognition (no Star Alliance benefits or equivalent) for high-revenue flyers. So, if you buy a lot of business- and first-class tickets on Singapore, you can receive PPS credit even if you accrue the miles for the flight in another Star Alliance program, like Miles & More or ANA Mileage Club. Obviously, this is only relevant if you are someone who buys a lot of premium cabin tickets on Singapore.

Special Promotions

The rare and beautiful double-dip promotion was sometimes seen when a new partnership was announced, but may be extinct.

In the craziest example in my lifetime, to promote their partnership after their failed merger, United and US Airways announced in 2002 that passengers would earn triple miles in both programs for flights on the US Airways Shuttle. Since at the time both airlines recognized a 500-mile minimum on flights, a single DCA-LGA round trip would earn you 3,000 Mileage Plus miles and 3,000 Dividend Miles before elite bonuses. These days, that round trip is worth a scant 428 miles for a non-elite… and I take Amtrak.

No, you’re not allowed to do that, and neither is it (usually) possible in practice. AAdvantage T&C:

Mileage credit cannot be earned for the same flight, hotel stay, or
car rental in more than one of the following programs: the AAdvantage
program or any other loyalty program in which American Airlines
participates.

If the airlines are in the same alliance, their systems are hooked up to each other, and will probably spot and reject an attempt to claim miles twice for the same flight. If the miles don’t register automatically, you’ll usually be required to send in your original boarding card stub, which also makes it difficult to double-dip.

Also, depending on the airline T&Cs, this could be considered fraud and they might go so far as terminating your account and your miles along with it if they judge that you’ve been doing this repeatedly and on purpose.

All that said, the systems that process frequent flyer miles are far from infallible. I was once upgraded halfway through a multi-leg flight in economy, and when the miles didn’t show up, I sent in my business class boarding stub. I ended up with two credits for the same flight on the same day, once in economy, once in business. <shrug>