In The Netherlands, in the nicer neighborhoods we have something called a ‘buurtbieb’ aka a ‘neighborhood libraries’, which is a weatherproof cabinet where people can put surplus books that other people in the neighborhood can borrow.
Of course you could take all the books or use the cabinet to store candy, but why would you?
(The name is a little weird, because regular libraries are also free...)
Private libraries (mine for example) are not free, as in beer or otherwise.
People tend to call their personal library a "book collection" or the like, but it's a library, in just the same way that a Little Free Library is.
So most people who read have at least a small private library, whether they think of it in those terms or not.
Furthermore, public libraries are not necessarily free. In America they virtually are all; fees only for late returns. But this is not globally true; in some parts of the world, libraries open to the public charge a fee for checking out books, or even require a fee for entry.
It's a literal translation of "neighbourhood library", it alliterates, and it sounds cute. (Keep the "beep" part short for that).
Good luck getting a jury to enforce the trademark.