If Airbus Beluga can fly with two engines, why doesn't Boeing 747 and Airbus A380 fly with two engines?

 It is the difference between weight and volume. The Beluga can carry huge components, but they are not very heavy. The Beliga is designed to carry oddly shaped, and therefore hard to carry components, specifically, wings and fuselage sections. The thing about the Beliguas is they exist to move big and awkwardly shaped components around Europe, particularly ferrying wings from the UK to France.

There's much more empty space inside a Beluga than there would be on the other two aircraft types. The Boeing 747 was designed in a time when engines were weaker than they are today. This airframe was designed back in the 1960s. Back then engine reliability was at nothing like the exceptionally high level we are used to in the present day. Both the B-747 and A-380 as well as the A-340 have 4 engines, not so much for the extra power, but in their early design days, you could not fly over oceans with 2 engines for more than about 60 minutes. When the 747 was first designed and built, engines in the 100,000 pound thrust capacity didn't exist, nor was anything even close to that available.

The Airbus A380 on the other hand, does need 4 engines. The A380 needs 280,000lbs of thrust. So as a twin-jet it would require two 140,000lb thrust engines. Even today there is no engine either in service or proposed that would deliver that much power. Due to its weight it needs powerful engines, but also because it's lower to the ground, the engines needed to be smaller.

The range and the MTOW needed is not as high as an A380 or B747 require, therefore two engines are fine for the Beluga. You cannot swap out 2 engines onto a 4 engine plane without re-engineering virtually every part of it. This would be too costly to even examine.

Was it a mistake for Airbus to build the A380?

 I think it was NOT a mistake to begin the program, but it was a mistake to finish it.

None of us have magic crystal balls that can reliably predict the future. Airbus took a gamble on a “super 747” and initially it looked like a great idea. The Russians were also developing something similar, though they were hamstrung by the fact that their economy was in the toilet after the USSR collapsed (though on hindsight, they are probably grateful that they didn’t go through).

Airlines and airports were positive about the airplane. Sure, it was going to be bigger than anything else, but many airports were willing to accommodate it. So, why not? As long as air travel was the same as before, the A380 would be a serious competitor to the 747.

What people didn’t realize was that hi-bypass turbofan engines matured. They give enormous amounts of thrust while minimizing fuel consumption. And they’re also deemed safe enough that you can fly long distances (across the ocean, for example) with just two engines—something unthinkable before.

This piece of technology pretty much killed the rationale for the 747 and A380. Why fly to hub airports like JFK, Dallas-Fort Worth, LAX, and Heathrow and take another flight to your destination when you can just fly straight to where you need to be?

At this point, it would have been wise to cancel the program or perhaps turn it into something else like a cargo freighter. But Airbus went with it and by the time it was launched, it was already on thin ice. If it was launched in the 80s or even 90s, the A380 might have enjoyed success.