As documented in many places, older versions didnt allow to have multiple databases in single instance. As in have separate /db/data location per database.
Based on this comment,
Neo4j 2.2 will be a first step in that direction.
I currently have Neo4j 2.3.3. Does the newer versions support this feature yet?
Not yet. Best way to manage multiple databases side by side on the same machine is using containers such as Docker - you can pick up the latest GA version of Neo4j on DockerHub. If you look at the latest 3.0 milestone (beta), you'll see a number of improvements, including changes to directory structure & config, to make this easier.