In theory, the absorbance of a sample is the same regardless of the spectrometer or Colorimeter used to measure it (as long as that exact sample is measured under the same concentration and pathlength conditions). Unfortunately, theory is not always what is observed in practice.
The main reason absorbance values of a given sample will vary is the nature of optical instrumentation. There are many different types of spectrophotometers, each of which use different light sources, detectors, and optics, including diffraction gratings or prisms. These differences in devices result in differences between absorbance values of the sample.
You also cannot assume that two devices from the same manufacturer will give you the exact same absorbance readings for the same sample. However, you should be able to assume that the absorbance measured from each device should be relatively close. The optics may be the same but the alignment of the optics will vary slightly. So two different Colorimeters should give you readings that are relatively close to each other, but they will not be exactly the same.
In general, the greater the cost of the instrument, the better the quality of optics and alignment used. This results in greater accuracy, reliability, and precision of the measurement; however, they still will not be exactly the same.
For best results, the same spectrometer or Colorimeter should be used for all measurements in any given experiment.