SHA-2, including SHA-256, is constructed using a Davies–Meyer compression function. That compression function starts with a block cipher - so an object like AES, but with wider keys and wider block size. For SHA-2 this block cipher is called SHACAL-2.

Now what we're seeing here is an attack on SHA-2 assuming a very, very significant degradation in SHACAL-2, where we run far fewer rounds than assumed in the standard. This is your typical cryptoanalytical result, interesting, but it is very very far from showing that "SHA-2 is broken".

As a side note I once estimated that the Bitcoin network is likely to produce a collision in SHA-256 sometime in 2050s, assuming the current rate of growth of the hash throughput. Of course that's a big assumption, and also nobody will notice the collision, as nobody is saving all those past hashes :)

Another side note - if you're interested in learning about hash functions then I recommend looking into SHA-3. Not because it's newer and shinier, but because I think it is actually the easiest to understand. It has a very clever design.