"Connect to a database" means a lot of things. If you're looking for an IDE feature that allows you to connect through the IDE and issue SQL commands and see results, I would check the IntelliJ resources. There certainly are plug-ins to do that for the Eclipse IDE, and I use them all the time.
If you are developing an application that is supposed to connect to a database, the IDE isn't responsible for that. And apps developed in the IntelliJ IDE can very definitely connect themselves to databases.
Many databases come with one or more "control panel" applications. For example, Oracle's SQL*PLUS, DB2's "db2" console command, PostgreSQL's "psql" console command, MySql's "mysql" command. There are often GUI DBMS control apps as well. And web-based ones such as the database plugins for WebMin.
It all depends on what database you are looking to talk to and how you prefer to do it.
Regardless, this is the
JSF forum and JSF doesn't have any DBMS features. It gets its database services the same way that plain webapps do. So I'm moving this question to the IDEs forum.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.