You've answered your own question, with a good choice of words!
You can put projects into categories. Log in as an admin, and go to Admin -> Projects -> Project categories, and add some. Then go to each project's admin section and select a category for them.
I'm afraid it is a single mapping, you can't put a project in more than one category, but it does the basics well. (I'd like to see project labels myself, so I can put several labels on projects and search by them)
The category does not do much itself, there's no effect on config. But you can run JQL like "Category = X" to find issues across a group of projects, which is sometimes more useful than you expect.
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