While User eXperience testing and design is generally associated with web sites and applications, there is no reason to take such a limited approach to the term and its underlying theory. Everything is subject to UX.
For instance, I just moved into an apartment with some baffling eccentricities. We have one long hallway with two overhead lights as well as two corresponding switches, but here's the catch: the lights turn on together and only if both switches are up. Thus it's easy to have the switches fall out of sync, particularly with two people on different sides of the apartment, leading to many journeys down a dark gauntlet. This is a usability problem: it is not as if the electricians could not have hooked the lights up otherwise. They chose not to, presumably without testing their audience or considering their decisions' ramifications.
Libraries are full of analogous design errors. Sometimes they are beyond the librarians control—such as when bathrooms are hundreds of feet from the most populous rooms—but many times improvements can be made. Testing our own services from a user's perspective is essential. Try out your chat reference service, or attempt to forget everything you know about classification and then locate a book on the shelf. Better yet; watch actual users do the same. It is trivial to find takeaways and small improvements quickly add up over time.
A concrete library example: I have spent a lot of time answering chat reference questions, but I have also asked a few. The chat widget my former library used cannot be resized, so users are limited to very short lines. I found two aspects of the experience distinctly cumbersome: copy-pasting full-length links that break across lines and catching up with consecutive multiple-line messages from reference staff. It is easy to convert these annoyances into praxis: shorten every link (preferably with a service such as bit.ly that provides analytics) and enforce a hard character limit on your continuous messages. I must admit I often fail to adhere to these simple maxims. In my perfect world, pop-up messages would alert reference staff when they were flooding users with too many words or using library jargon.
This is not meant to pick on libraries; we can see usability problems almost everywhere. Next time you visit the DMV, the Post Office, Wal-Mart, or the movie theater, ask yourself "What is frustrating about this experience? Where are they failing to meet my (reasonable) expectations?" Perhaps you can come away with a few lessons for your library as well.
