Is the measurement accuracy of a test instrument consistently the same?
Accuracy of an instrument depends on many factors. While some may remain (relatively) constant, others can drift in unexpected ways.

For a DMM, the greatest source of error is likely the voltage reference -- this basically is the value against which most measurements are compared. When measuring voltages on a single range, this may vary slightly with temperature, but in a good quality ('name brand') DMM, the variation will be small. If you switch ranges (e.g. from the 10 V to the 100 V range), variations in the resistor divider will be different, and so your error will not be the same.

In current measurement, AC voltage measurement and resistance measurement ranges, there are other elements contributing to the errors, and so you cannot compare errors on the DCV range with errors on those ranges. Usually AC range errors are non-linear (significantly greater at low input levels), and so even the linearity isn't necessarily constant.

In fact for low inputs, DC offset voltages also contribute to errors -- and this means that results are not always off by a fixed percentage, but also by a constant value.

In general, for a fixed DCV range, if you verify that the meter reads '0.00' with the leads shorted, the gain error (e.g. 102 % of the true reading) will be quite constant. Same for oscilloscopes inn unchanging scale.