Oddly I can trace my interest in databases back to playing Championship Manager in the early-mid nineties. For those not in the know, it's a football management simulator that, apart from the actual game mechanics, is basically a rdbms.
I made my own in FileMaker Pro for a school project. They were the days.
From what I understand (I'm just starting to look into this myself as a potential career option), database design and other technical skills like scripting and web design are becoming an increasingly important part of library science. There's a new-ish degree, "Masters in Library and Information Science", that combines both library and information knowledge like this.
There's plenty of open source library management tools such as Evergreen[1] or Invenio[2] for more standard data (like book metadata) but it's still helpful regardless to understand how databases work, what fields might be useful, how normalization works, etc. to better organize and understand the data in a library.
When I got my MLIS in 2007, it was not a very technically rigorous program, and the technical classes were only meant to introduce non-technical people to systems they might eventually have to talk about in a meeting, but not work with directly.
The people I graduated with mostly went into libraries (naturally), or else went into Product Management, UX, or Information Architecture. If you want to have a contributing technical role in your future position, I urge you to ask very specific questions about curriculum of the program you consider applying to!
I worked at a place that hired librarians to manage the data in the database and keep it clean. It was pretty wild, and when I've suggested this at other places I got blank stares.
The two terms are not wholly synonyms, even though they are closely related. The discipline is often called Library and Information Science or LIS for short.
I personally call it Documentation because I like Otlet.
I made my own in FileMaker Pro for a school project. They were the days.
https://en.wikipedia.org/wiki/Championship_Manager