No they aren't? RNNs have state that gets modified as time goes on. The RNN has to learn what is important to save as state, and how to modify it in response to different inputs. There is no explicit time-stamping.
However, that doesn't actually eliminate the axis/dimension. Eliminating timestamps only makes the dimension a unitless scalar (IOW 'time' tautologically increments at a 'rate' of 'one frame per frame').