What is this transformer-like component, and why is it not really connected?
It's supposed to look like a passive PFC choke. Real PFC chokes were common in the early-mid 2000s. Those were replaced by active and lighter counterparts on more recent power supplies. 

Worst passive PFC knock-off I heard about from my friends in China was that the customers associated heavy weight with PFC chokes, so the manufacturers put a large stone in each non-PFC power supply and marked it with PFC. Then the customers complained that there was a rattle from the power supply to they "improved" quality by wrapping the stone in newspaper to cushion it. 

More examples from the interwebz: