How can my case activities show up on an organisations profile?
I think by "tag" you mean put their name in the "reported by" field or something like that. Case activities don't appear in "non-case" contexts. This was originally a design goal to both keep case data more private and force people to keep case data inside cases instead of a mix of places.

So the short answer is you need to go to the Manage Case screen to see case activities.