It doesn't have to be, but both are, at some level of abstraction, switches that integrate multiple weighted inputs and set an output level that's used as input for other switches.
If this is what we expect, the analogy holds and the implementation details don't matter.
But there is a fundamental difference that breaks the analogy: the "function" that the neuron uses to set its output based on its input is not a function at all, it is a stateful mechanism.
That is, for the same input at different times, a real neuron will have different outputs; and its own ouptus may change its state. In contrast, the memristor, once programmed, always applies the same function to its inputs. Even if it can be reprogrammed with a different weighting, it can't do so based on its own output - at least not with any current neural network architechture.
I never said the "artificial neuron" would be stateless. Natural ones have states and, if we model artificial ones as stateless it's just because they are easier to model that way. In that sense, an artificial neural network is an exercise of how little "neuron-ness" one needs to make a useful neural network.
