Just because neurogenesis exists, it most definitely does not follow that "every single cell in the body of a plant or animal is replaced every few months", which is what you said.
Can live up to 50 years. OK. What happens when they do not? Does the body not regenerate them? Various kinds of stress can cause them to not live that long. It is not as black and white.
Adult neurogenesis or brain plasticity is a fact. Skin cells, blood cells and all types of cells have a finite lifespan - a few days to a few months.