Organizing Postgres Databases
Just type it into the "Group" field. Besides the listed options you can type a new group name freely.

It's a setting of pgAdmin, completely independent of the database itself and not stored in the database.

The handling is flexible, but renaming a bunch of "servers" can get tedious.

As to your comment: I repeat what I already wrote in my last answer: 


  Basically, each connection allows access to a whole database cluster


But if you follow my instructions over there, you connect to the right database automatically. I suggest, you read the manual about Managing Databases for basic understanding.