The study itself does not surprise me. There are a lot of psychologists, anthropologists, and sociologists who contend that conservatives and liberals have quite different patterns of thought. I decided to do some due diligence, and the study is published in a peer reviewed journal, Ryota Kanai is a respectable scientist from what I can see, and the sample size isn't too small (90 isn't that bad actually). Where the bullshit comes in, is the description of the structures of the brain.
The Amygdala is not about "fear and anxiety." It's a structure of the brain that is in part used to store emotional memories, primarily those that are associated with fear, anxiety, or conditioning. This is both a good thing and a bad thing; it can lead to the formation of biases which are baseless in reality (racism, sexism, irrational fear of certain things like socialism) or it can lead to enhanced memories about what does and does not work as a response to dangerous situations. The Amygdala also has an effect on memory modulation, which is the process of transcribing things form short term memory to long term. Basically the effect is that the stronger the emotional tie to that event, the more it is emphasized. Again, this can be a good thing and a bad thing: it's a mechanism of confirmation bias and a way of ensuring that important things about the past are not forgotten (which is one of the key roles of conservatism).
The ACC isn't as well understood. There's a lot of contention about what it does. Some people believe it plays a role in feeling negativity about one's own errors, but others think that's inaccurate and it's actually centered around reward-based condition. If you take the latter view, a whole lot of politics can be boiled down to positive reinforcement versus negative reinforcement. Unfortunately that's much too simple of a view.
Bottom line is, we don't know enough. We don't know how these portions of the brain being emphasized effects their processing, what they actually do 100%, and how it affects us. I wouldn't get too worked up about this, but it does seem to be solid science.