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).

custom categories vs custom tags

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