Here’s a gamut plot of my monitor (the coloured outline) compared to Adobe RGB (the dotted grey outline):
This is a 2D representation, so it doesn’t show you the full picture, but descriptively my monitor covers 97.9% of sRGB and 97.3% of Adobe RGB. However, its full gamut volume is 148% of sRGB and 102 of Adobe RGB, so it exceeds both colour spaces in some hues (yellows/reds), but it doesn’t contain them entirely (magentas/blues). That’s why it’s not optimal to profile my monitor to either sRGB or Adobe RGB but rather to its native gamut. This is the profile which is picked up by all colour-managed applications.