Custom/separate categories for custom post type

How can I achieve a separate “data space” for the categories of my custom post type?
I don’t want the general post categories to appear in my custom post type and vice versa.

The register_taxonomy function seems to add a “tagging system” (bottom panel in the screen) to the post type but not the same small window the “real” categories have (upper panel in the screen).

You can use register_taxonomy to create a category-like custom taxonomy using the argument ‘hierarchical’ set to true.

In Codex example see the first taxonomy registered (“Genres”).