Working on other stuff (side projects and activities) without permission from the professor and department

I was thinking that is the norm in academia, however in my new lab people seem to work only on their project and don’t do anything else. While I am not prohibited to work on these things, I am uneasy and puzzled to continue, but also I am not sure if bringing this topic to my professor is the best idea.

I agree that the norm in academia is, and should be, that staff on all levels need to have sufficient freedom to pursue their own interests in addition or within the funded project(s). As a postdoc, this is particularly troubling, because you need to use your postdoc time to diversify your research themes, skills, and collaborative network. Basically, if you are a postdoc and sitting in your office all day working on a single project, you are doing it wrong (and hurting your career big time).

I don't think that sneaking a little side project here and there by your professor is sufficient. As a postdoc, you should be able to dedicate significant work time towards progressing your own career. I suggest you bring this up with your professor, and, if they don't appear to understand or accept your concerns, I would start brushing off the CV and get back onto the job market.