That's why the meme works. It's not because water autoionizes; it's because water is amphoteric, meaning it can act as either a Brønsted-Lowry acid or BL base depending on what what it's reacting with. Put water with ammonia, and water acts as an acid. Put water with acetic acid, and it acts as a base
Source: I teach college chemistry
Interestingly enough, in other solvents a neutral pH is going to be a different value. IIRC, ammonia has an autoionization constant of 10^-30, so a neutral pH would be 15