[Samsung Galaxy S23 Ultra] dl12345:
Yes, my monitor is wide-gamut. I'm not 100% sure exactly what you're asking, although I'll try to give a short explanation on colour management. This info is not specific to phones. It's general colour management information. There are lots of resources on the net if you wish to learn more. Some of the info below is guesswork, since Samsung's colour management is a bit of a black box.

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Android 13 is colour managed. That is to say, ostensibly it has all these components for a fully colour managed workflow, although many apps are still not colour managed.

On a properly calibrated and profiled desktop system, any photo should display using the same colours and look identical to the way the same image displays on a different fully calibrated and profiled monitor. For example, a calibrated and profiled AdobeRGB monitor should display an identical sRGB colourspace image to a properly calibrated and profiled sRGB monitor. This is how it should be.

Take, for example, the RGB value RGB(255,0,0). On a wide-gamut monitor, this maximal red value will be a deeper shade of red than on a smaller gamut sRGB monitor. So when no colour management is present, this would mean that instead of displaying the correct sRGB red primary colour, a wide-gamut monitor would display its version of the red primary, which is way more saturated (deeper) than the sRGB red primary. Colour management ensures that the sRGB red primary will map to the equivalent red hue in the wider monitor gamut and thus display correctly.

On the S23U, the display device profile shipped with the device appears to be non-standard and deliberately displays more saturated colours - I assume this is precisely what is intended by the "Vivid" setting in Display Settings. It looks almost as if no mapping is being applied between the image's sRGB colourspace and the wider display gamut, resulting in more saturated colours. Why Samsung chooses to do this, I don't know - it's probably because these cartoonish looking colours are seen as a desirable marketing feature. Without access to the source code or detailed technical documentation on Samsung's colour management implementation, it's just guesswork as to what they're doing. It's why the icons look more saturated in vivid mode: their sRGB colour value is being displayed in a wider-gamut space, resulting in a more saturated colour.

To partially fix this, setting sRGB on in Developer Settings is supposed to force the display's gamut to emulate sRGB when displaying pictures and so the image colours display as intended rather than in the over-saturated form that Samsung thinks consumers want (although it seems to be less than perfect in its emulation of sRGB to be honest, as it still looks a little more saturated than when using the Natural colour scheme). One would reasonably expect that pictures displayed when Picture Mode sRGB is on with Vivid mode to be the same as how it would look if you set the display mode to Natural, although it's not a 100% match (it's close). This Picture Mode setting is essentially a picture-specific sRGB mode as opposed to the system-wide natural display mode that forces the display gamut to emulate sRGB at all times.

This Picture Mode setting apparently has no effect when watching videos. It only affects pictures. As an aside, having watched several videos encoded in different colour spaces in the different S23U display modes, video colourspace mapping appears to work properly (ie. no over-saturation). Most Youtube content is encoded to REC.709 which has a similar gamut to sRGB (REC.2020 is used for HDR and is much wider gamut). I've watched a combination of REC.709 and REC.2020 videos on the S23U while running the same video on my desktop at the same time and the videos look identical on the S23U. In fact, video colour reproduction looks remarkably accurate.