While they investigated new topics (i.e. economics), their methodologies were 100% within the standard practice of social and cognitive psychology at the time.

They were then awarded the Nobel Prize for this work. Now, not everyone in the field came from psychology, but a lot did, and the methods used in behavioural economics are essentially indistinguishable from those used in psychology.

For references, Thinking Fast and Slow is probably the best source, as one of the original authors describes their research program.

Also, modern economics use some heavy mathematical tools, yet nobody says its a branch of mathematics.

And lastly, psychology is in my opinion one of the most important branches of science right now.

I agree with you on psychology's importance, but that's probably because I am a psychologist :)

Macro investing literature is what you should read if you want to understand the economy. It's less focused on high level concepts and more on specific data points and how they relate to and impact one another.