How to make colours on one screen look the same as another

If you are talking about lab-critical calibration (that is, the colours on one monitor need to exactly match the colours on another, and both need to match an external reference as closely as possible) then a hardware colorimeter (with its own appropriate software and test targets) is the only solution. Software solutions can only get you so far.

The technique you described is a common software-only solution, but it's only for setting the gamma curves on a single device. There is no control over the absolute brightness and contrast; you are merely ensuring that solid colours match their dithered equivalents. That's usually done after setting the brightness and contrast so that black is as black as it can be and white is as white as it can be, but you can still distinguish not-quite-black from black and not-quite-white from white. Each monitor, then, will be optimized for its own maximum colour gamut, but it will not necessarily match any other monitor in the shop (even monitors that are the same make and model will show some variation due to manufacturing tolerances and age/use). A hardware colorimeter will (usually) generate a custom colour profile for the device under test as it is at the time of testing, and there is generally and end-to-end solution built into the product (so your scanner, printer, and monitor are all as closely matched as they can be).

You will never get to an absolute end-to-end match in a complete system, but hardware will get you as close as you can get. Software alone can only get you to a local maximum for the device it's calibrating, independent of any other device.

What you need to investigate are color profiles.

Wikipedia has some good articles on this:

https://en.wikipedia.org/wiki/Color_management

https://en.wikipedia.org/wiki/ICC_profile

The basic thing you need is the color profile of the display on which the color was seen. Then, with the color profile of display #2, you can take the original color and convert it into a color that will look as close as possible (depends on what colors the display device can actually represent).

Color profiles are platform independent and many modern frameworks support them directly.

You may be interested in reading about how Apple has dealt with this issue:

Color Programming Topics

https://developer.apple.com/library/archive/documentation/Cocoa/Conceptual/DrawColor/DrawColor.html

You'd have to allow or ask the individual users to calibrate their monitors. But there's enough variation across monitors - particularly between models and brands - that trying to implement a "silver bullet" solution is basically impossible.