Wrong. There could be tons of different things going on inside, absolutely indistinguishable from outside, which only sees mass, electric charge and angular momentum.

Nitpick: this is only true for a stationary black hole. If you throw something sufficiently big in, you would expect the shape of the horizon to change and bulge a bit, until it settles down into a stationary state for a larger black hole. You are of course correct that this does not allow anything inside to send a signal to the outside.