Q:

How do I build a related list against a Files Library (in Lightning)

I want in the related tab of a record to have a Library related list shows all the files in a library? Is that possible?

A:

Based on what I understand is that you want to display the "Libraries" section available on the "File" tab in a Tab on a record details. As of today, there's no standard out of the box way to do this. I don't at least see any component available on Lightning App Builder, which you can utilize for this purpose.
I would think this to be expected behavior considering that Libraries are a way to organize your Files which is not associated to any particular record. And that what you see on a record is a specific file associated to that record.
If at all you require one, the only option I see is that you need to create a custom lightning component and use ContentWorkspace to be able to pull all libraries and display that accordingly.

