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No. Document profile and monitor profile are two entirely different things. Don't mix them up! And above all, don't ever set them to one and the same. That defeats the whole purpose, it effectively disables and bypasses the whole color management chain that ensures your files are correctly represented on screen.
The monitor does not have to match Adobe RGB or anything else. It has its own native color space. In a color managed display path, the RGB numbers are converted/remapped from the document color space into the monitor color space.
Any color management operation always requires two profiles, a source and a destination. One is converted into the other. A single profile, alone, does nothing. It's like one hand clapping.
Don't change anything in Color Settings! The embedded document profile (from ACR or anywhere else) should always override the working space. The monitor profile, OTOH, is set up at system level, and Photoshop gets it from the OS and uses it in a standard profile conversion, under the covers, without any user intervention.
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You need to separate the monitor profile from the image profile, you appear to be confusing the two, and let the color management sort out the conversion between the two.
1. Your monitor.
The profile is set in your system (not in Photoshop). Ideally you should get a hardware calibration and profile device and use that to get the actual characteristics of your specific monitor. If you are producing images for fashion catalogues where colour accuracy is important this really is an essential.
Until you have a device then in the short term set the monitor profile in the operating system to sRGB or to Adobe RGB depending on how you have switched your monitor.
In Windows 10 the profile is set in Settings > System > Display >Display Adapter Properties > Color Management
2. Your Photoshop color settings
These fall into two - first the working space defaults for creating a new image. You are safe setting RGB to Adobe RGB (1998) but CMYK depends on where you live in the world. That is why you will see some presets for different uses and parts of the world.
Second - Color management policies. Always have these set to Preserve Embedded Profiles. That way the color management will sort out any conversion you need between color spaces.
4. When exporting an image for use on the web - always tick the options to "Convert to sRGB" and "Embed Color Profile"
Once set up , let the color management do it's stuff. No need to mess around with settings.
Daniele - D.Fosse answered whilst I was typing and is our color management expert . Follow his advice and you will be fine.