In simplistic terms, 'calibration' consists of two parts - calibration itself and profiling. The two go together to make your final 'calibrated' output. People tend to use the terms interchangeably, perhaps as they are always done as a pair, the distinction blurs.

The actual calibration part tends to be relatively simple when done from computers. You set the display to default, then tweak brightness, contrast & perhaps individual RGB levels to reach a desired start point.
That's all the information that is held in the display. The rest is done in the computer & stored there.

This is the profiling step.
This takes values sent from the computer via the graphics card to the display, then measures the difference between the colour sent & the colour received. It repeats this process for many dozens or hundreds of colour samples. It then compensates each as an offset between input & output values, as measured by the colorimeter. It will re-check some of these calculations before presenting you with a new colour profile file to store & use, usually as an .icc file.
This profile then intercepts at OS or application level so all output arrives at the display as intended. [Different OSes seem to handle this in slightly different ways, the details of which I am not fully conversant with].

In theory, you could copy this .icc file over & install to another OS using the same hardware. In practise… I've never tried it. I'd be inclined - weighing up the overall time saved - to re-profile in each OS.