Actually, #E58C4E, if you mean the web color, is defined to be in sRGB. However, if you didn't mean that particular convention but rather "red:229, green:140, blue:78", it's a different matter, because the extremes (the "primaries") of each channel are different in different color spaces, so those numbers actually do represent something different in each space. (Arguably, that's what a color space is.)

The monitor has its own primaries (the native color space of the monitor), and a color managed system translates between a known profile for a given image and the native output. If the color space of the original isn't known, there's no way to guess at the proper translation.

So, another way of asking the question might be: why aren't colors from un-color-managed-applications just assumed to be sRGB and translated on the fly? From the monitor's point of view, that's easy: it has no idea about which pixel came from which application, so it wouldn't know what to translate and what to leave alone. The graphics driver level has the same basic problem. Going from the other side, un-aware applications can't do it, because they're unaware. So, that leaves the operating system's display manager to do it, and that's hard.

So here we are.