How can they look the same? One monitor is set to 1.8 and the other to 2.2 - if everything else remains equal they should look like this: http://imageshack.us/a/img407/6302/gammadifference.jpg.
The key to the question is that it says view an image with "an embedded profile in a color managed application" which means the monitor profile is used to determine what standard the monitor is set for so the viewing application can
change the image data to match the requirements of the embedded profile. Since in either case it is the embedded profile which determines the final appearance, the usefulness of the two different monitor profiles is as an accurate starting point when adjusting the image data. You don't end up viewing image data from the original file, and instead see adjusted data edited to match the embedded profile to the selected monitor.
Of course if the embedded profile does not produce an image that matches your
printer output (for example because it was constructed for use with a different paper), then the displayed image will not be what you get when the image is printed.
That is one reason that I do not prefer to have color managed by each application, and prefer to ignore embedded profiles by default. Those do work for some people's workflow, but is awkward for the work that I typically do. I usually want to see
everything in one partricular manner regardless of the viewing program or the embedded profile. For example when working on images for web display everything looks approximately correct for an sRGB monitor set for 6500K or higher (many default to above 9000K), with gamma 2.2 and with slightly excessive brightness for the environment. That means it won't be far off for most web viewers. But when working on print jobs, one of several other monitor profiles are more appropriate.
I don't work to the lowest common denominator of color management, and instead fine tune color management to the specific workflow being used, which is optimized for more flexibility in areas of more significance for time or other efficiency.