Fun fact, on average most (not all though) of the cells in your body are brand new after 7 years. When do you stop being you and take a new name?
More relevantly, I don't think neurons are replaced. There must be some material churn in the atoms and molecules that make them up, but even then different for different molecules - e.g. I don't know how much of our DNA molecules get replaced over a lifespan from the repair or other mechanisms.
Tattoos however, IIUC, sort of "float" between cells, and as those cells are replaced one-by-one the ink is kept in place by the surrounding cells that are still there.
Answered by Kurzgesagt: https://www.youtube.com/watch?v=nGggU-Cxhv0