The answer about "Chained Libraries" is probably the best answer and that it was ultimately just an aesthetic choice to show parallels with our own history, since it doesn't appear there are books attached to chains.
But as an aesthetic choice one can argue that it plays with themes. Game of Thrones is a cycle cosmology story, meaning the reason the metaphysical aspects are happening, including the White Walkers coming for the living, is in part a metaphor for people having to "deal" with the indiscretions of past generations (the sins of the father = karma). It's why Martin went to great length to write "histories" and that 'songs and seasons' are additional motifs to solidify this idea, as knowledge is power and power is an opportunity to control and/or gain freedom or liberation.
"I will not stop the wheel. I will BREAK the wheel." - Daenarys Targeryen
On Dany's side of the story, the idea of freeing people (one of her monikers is the "Breaker of Chains") is obvious in her over all role, but it's that her purpose is essentially pointed in the wrong direction with Essos plots being a microcosm of the macrocosm, as it is really the freedom to live (and hopefully live better by abolishing a monarchy government) that is the purpose of our story, as defeating the White Walkers once and for all, is how the cycle will "break" into a new (ie: The Law of Averages). So the Library itself is another kind of display of this idea.
However, there could be other historic significance to the Library having chains. Oldstown and certain parts of the Citadel have other histories, including that Pirate Lord once claimed a portion of it home and picked off ships when they came in. And although it is disputed, there once was a king also whom may have built the towers and had an assortment of magical persons people come to his service.So it's categorically possible that Library was once something else and the chains belonged to it, more than library.
Note: White Walkers had chains to get Viserion's body out of the water was also a physical feature in the season.