You wouldn't set or select your monitor profile anywhere in PS or in any other color managed program, even though it may be shown as an ICC option. It is a system wide environment that isn't set or selected anywhere else other than Display > Settings > Advanced > Color Management.
You could say that your monitor profile is basic room lighting. When you then bring an image into the room that requires Adobe RGB lighting, then according to the aRGB profile embedded in the image it will alter the lighting to be correct for that image. If you bring in an image with a sRGB embedded profile, the lighting will be altered to best display that image.
If you are using PS principally to publish images to Web, then you might want to set your working color space in PS to sRGB and make sure all images you bring into PS be sRGB (shoot sRGB) or convert them to sRGB if they are not. Then process and publish.
If you are mainly viewing and printing your images, then you may want to open your image in PS using it's original embedded color profile to get the maximum out of it.
The key to all of this is to be sure your monitor is correctly calibrated and profiled, preferably with a good colorimeter. This involves correcting your monitor with your monitor controls, changing settings in your graphic card's Look Up Tables (LUT) and writing a description of your monitor's capabilities, characteristics and settings for Windows and color managed programs to refer to.
If your monitor is not correctly calibrated, it will have absolutely no direct affect on the other output forms. Your display can look terrible, but at the same time your prints are perfect. If you were then to try to "correct" your perfect images in PS to look right on screen, your prints would come out a four-letter pile.
I view each output form, screen, print, web, etc. as quite seperate, but having a common hub. If I get each individually correct, they will work correctly with each other over the hub.
Alternative non-technical description, hope it helps...