Word or phrase to describe how frequently a value is measured

The name for the frequency the samples are taken is Sampling Rate (not specific to audio only). It is measured in samples-per-second or in Hz (which is 1/s). The time between two samples is called sampling period, and is given in units of time. So in your case the period would be 1ms and the frequency 1/1ms=1kHz. Or in terms of rate it's 1000 samples per second.

I would call it the sampling period or measurement period. For short periods like 1 ms, specifying the sampling frequency is another option (1 kHz).

The word is "period". Could be used in the expression "sampling period", in your case.