…doesn’t it usually imply…something you have in common?

Yes. That you live in the same area. You have the same elected representatives, you use the same public services, you shop at the same stores, you live with the same stakes. That’s why people say you need to get involved in your community. Whether or not you acknowledge it, you’re part of it

Edit: to add, “community center” probably sounds old-fashioned to you, because of what they were talking about:

it’s hard to get anyone to be willing to be remotely friendly or willing to admit that they are destroying the social fabric of their own Communities.

Getting rid of places where you can engage with your community has been part of this effort. Community centers are run by local governments, and are often one of the first things conservatives try to cut