The chances you will ever get two different monitors, each on a completely different OS to look the same by eyeballing the color is pretty darn slim.
A monitor profiling solution is the the cheapest, and probably the most important piece of color management equipment you can buy. $250 for an (example) X-Rite i1 Display Pro is worth every penny.
The built-in Calibration function is literally only that - a calibration. From there it entirely guesses at a profile by doing a mathematical calculation based on an assumption. And even the calibration is a guess since you're doing all of it by eye instead of getting actual readings from the monitor.
With a profiling solution, you'd have the software included for both OS's. From there, it would be very easy to match the monitors to each other perfectly. Or at least, as near to perfect as possible depending on each panel's age and the control each allows.
Calibration is setting the brightness, gain, white point and black point. The latter two go together as your gray ramp. That is, do you want to use a 5000K neutral gray balance, or the ugly and very blue 6500K default?
Once the calibration part is done, a series of color patches are then read by the device to create a profile based on the completed calibration. A final profile is than saved and set as the default. The profile includes the LUT data so the calibration settings created for that profile are loaded at the same time the profile is chosen.
