How much independency does a postdoctoral researcher have?
I would like to ask your opinion on whether it is normal for postdoctoral researchers to have collaborations with other people as well?

Our opinions don’t matter. The professor in charge of your grant is considered your supervisor, and whether it’s okay for you to spend significant time on outside collaborations depends on, and only on, the professor’s opinion. It’s likely okay, and yes it is somewhat normal at least in some disciplines (e.g. math), but you should simply ask them.