Johnny Cash wasn't a darling of the music industry when the Folsom Prison concerts were recorded. He had a drug problem which made him unreliable at best, had been arrested for drug possession, and, he had to fight his record company to record the album at the prison. His career was not in great shape.

It seems reasonable to expect that the concerts might have tarnished his image, because who wants to be associated with prisons. But Cash had been playing these concerts starting back in the 50s, and the association and identification he had with the inmates actually became a part of his identity. He was the man in black, an Everyman who realized that if a few things in his life had gone a tiny bit different he probably would have ended up behind bars, too.

So, rather than harming his career, or garnering backlash, the Folsom Prison album has been credited as the turning point in his career. It was that album that solidified his public image and started the ball rolling in what was to become an amazing career. I don't think there is any way it could happen like that in today's climate, but his concern and affinity for those oppressed and downtrodden was seen as a Christian attitude and he was widely admired for it.

My primary source for this is the Johnny Cash at Folsom Prison documentary, which I have seen a few times. When confirming my recollections I pulled up this interview with Michael Streissguth who was a producer on the documentary.

To answer the second part of your question about whether he was unique playing prisons, he was not.

The Grateful Dead played San Quentin prison in 1967, and Terminal Island Correctional Facility in 1971.

The San Francisco band Crime played a show at San Quentin in the late 70s.

Best part is they dressed up as cops like they always did. There's a video somewhere online. Pretty funny and a killer show too