VGA has pretty good EGA-compatibiltiy, and "somewhat passable" CGA-compatibility.
(Why only "somewhat passable"? - no composite output; different refresh rates; some registers operate differently; tweaked CGA modes are not available; no option for text modes in CGA resolution; etc).
It isn't sufficient to test just one game and conclude that there is 100% compatibility. Prince of Persia might show CGA/EGA graphics just fine on a VGA card, but i can guarantee that games like Digger or Round 42 won't, as they use features that VGA doesn't emulate properly.
This doesn't have much to do with the choice of monitor, btw. The only incompatibilities I can think of that concern a monitor (LCD vs. CRT) are things that specifically rely on CRTC effects, like the overscan/border area, "copper bars" and other things that rely on how a CRT monitor physically operates, which means they won't work on an LCD, even when everything is purely VGA.