The transplant itself was done 48 years ago and the mother was 52 at the time (thus the kidney was already 52 yrs old at the time of transplant)
https://www.inc.com/lucia-auerbach/can-transplanted-heart-be...
like the donee dies, and the original kidney goes on to someone else
Organs can be transplanted twice but it's very rare and it's usually when the first recipient dies shortly after the transplant. It's called (quite obviously) retransplantation if you want to Google it.
All types of organs can live that long, there are 100 years old humans around. Probably many types of organs could live much longer, we don't know that, the humans having them die first.