Attachments luckily can use custom taxonomies – that is what you are looking for.
Basically, you need to create a custom taxonomy, and configure it to work with attachments only. Usually the best way to do it is programmatically, this guide can help with that.
For a simpler solution, this plugin could work, or maybe PODs, but I would avoid using bigger plugins like PODs (or even the smaller ones if possible), unless you need more of their features.