It's not a nickname. It's simply because the prison is located on Rue de la Santé (Health Street). Rue de la Santé was named that way because there was a hospital there from the 17th to the 19th century.
But got covered by Wikipédia "https://fr.wikipedia.org/wiki/Prison_de_la_Sant%C3%A9", so I did called it a nickname too.
I'm pretty sure it can be called a "Métonymie de lieu" but I just didn't want to insist about that, it feels a little pedantic.
Typical French conversation then!