Why is District 6 like an island?

I am assuming that you are referring not to District 6’s geographical features nor culture but rather to its citizens's isolationist tendencies.
More than any of the other districts, District 6 seems the most culturally disconnected from Panem. Its tributes either die quickly (like in the 74th games) or are dependent on morphling (later in the series), thus limiting their character development.
Despite transportation being District 6’s largest industry, Katniss suggests that its citizens do not travel much outside the boundaries of their territory. In this way, one could argue that District 6 conducts itself much like an island would because of its tendency to self-isolate.
The insular quality of the citizens from District 6 is also attributed to the widespread addiction to the synthetic painkiller known as morphling. A dependence on drugs feeds this tendency to isolate, thereby making District 6 like a metaphorical island, separate from the rest of Panem.