Many people would say that it cannot, a softer thing cannot penetrate a harder thing, like a knife cuts butter while butter does not cut holes to a knife. In a similar way, as the plane is a hollow vessel made out of relatively soft aluminium, it cannot cut through a solid structure made of much harder steel.
But it is not quite so.
When an aluminium plane hits a steel framed building with the speed of 260 m/s, the aluminium as the softer material is crushed and steel is damaged to a much smaller extent. It is exactly the same that what would happen if the building moving with 260 m/s would hit a standing plane; the plane is completely destructed, the building only slightly damaged. The same result you get if you ride a motorbike and crash with a truck. The motorbike is completely destroyed, the truck hardly at all.
But this small damage to the building can very well be that the mass of the aluminium plane (the mass is not only from aluminium but form fuel and payload) makes a hole or even goes through the building. It depends on the kinetic energy.
t The hollow vessel made of soft aluminium, the plane, had the mass something like m=100,000 kg and the speed something like v=260 m/s, so the kinetic energy E=0.5mv2 was about 3*109 J. The pressure to pierce steel is something like 400 MPa, that is 4*108 N/m2. Let us assume the plane should cut an area of 10 m2, probably it was less. So the force needed is 4*109 N. How do we go from force to work? Easily: if it takes some amount of work to make a hole to a steel plate of given thickness, then if we put two such plates on top of each other it takes twice the work. If we put N plates on to of each other, it takes N times the work. So, the work needed is the force times the thickness: we can think in a simplified way that the plate is made out of a pack of very think steel plates, we have to make a hole on each of them, that is: force times thickness.
On low levels the steel columns had the thickness of 10 cm, 0.1m, on upper floors lesss, but for this cover-of-an-envelope calculation, let the thickness be d=0.1m. Thus, if every layer of steel resists independently the work is W=Fd. (Actually it should be less as the steel layers are not independent, but let it be so.) Then the work to pierce this steel is 4*108 J and we have 3*109 J of kinetic energy. It should go through easily. It makes a hole.
But armour piercing bullets have sharp ends and are made of material harder than steel, why is that?
It is because they have so small mass that they hardly manage to pierce the armour at all and they must be made from very hard material so that their head does not flatten and the area that they have to pierce does not become larger. The energy needed depends linerly on the area that the bullet must pierce, so it is much better if the bullet stays sharp and does not flatten. It does not matter for the plane hitting the skyscraper because there we had more kinetic energy than needed but bullets do not have so much energy,
It is because they are so small. Bullets that can pierce steel are typically light, m=30 g to 300 g and the speeds maybe around v=1000 m/s. That means that v2 of 4 times higher speed is 16 times higher than for the plane, but the mass is 3*105-3*106 times smaller than the mass of the plane. The kinetic enegry E=0.5mv2 is so much smaller than in the case of the plane and the skyscraper that these bullets need every help from tougher material (like tungsten) to pierce steel armor. Their head must not flatten if they are to penetrate steel. But if you just have enough mass, then even the speed of the plane gives very high energy, and it will go through. It does not matter if the plane flattens as long as there is enough energy.
The plane will be totally crushed, but the mass does not vanish to the air. The mass continues forward because momentum is conserved, and it has this energy that the building must mostly absorb as only some kinetic energy can be bounced back in such a collision. Indeed, if the mass makes a hole, the plane vanishes to the hole and nothing bounces back.
There are also people who show a video of a plane penetrating the WTC buildings and claim that the plane should have at least slowed down. Well, it must have slowed down as it did some work breaking steel structures, but not necessarily so much that you would notice it.
As the difference between the work needed to make the hole and the kinetic energy of the plane is so large (the plane has about 10 times the energy needed to make a hole), the plane will not slow down in a visible manner but will cut the building like it was butter. The mass of the plane slows down a bit, but you cannot notice it in the video. If it uses 10% of its energy, then its velocity goes down by 5%. Let x be the decrease of the speed. Thus, the energy after breaking the steel it (v-x)2=v2-2xv+x2. It it takes 10% of the energy to break steel, we have 2x/v=0.1 and thus x/v=0.05, that is 5%. The speed of the plane decreases only 5%.
You cannot notice in the video if the velocity is 5% smaller.
What however is impossible is that one video shows the nose of the plane coming undamaged from the other side. The plane is totally damaged to compressed mass when it hits the building and cannot have a nose to come out from the other side. And this is why the videos shown may be fabricated. But the correct argument is not that softer material cannot pierce harder material. It can if it has enough kinetic energy.