Quoting from the Description box of Photoshop's color management settings for the RGB "Off" policy:
"When importing colors into a document, the numeric values of the colors take precedence over the color appearance."
An assignment will take place. Using your example, I verified this by pasting an sRGB rainbow gradient image into an Adobe RGB document with RGB policy set to Off. The rainbow gradient changed to the same appearance one would see if they had taken the original sRGB document and reassigned it the Adobe RGB profile.
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There is no “No Color Management” behavior in Photoshop despite the silly OFF policy. The RGB numbers are untagged and thus undefined but under the hood, Photoshop is still using its color management and Display Using Monitor Compensation architecture. The RGB working space you set in the Color Settings is what is used for the assumption of the scale ( color space) of the untagged data. Same with CMYK etc. All you have to do is open an untagged RGB doc, go into Color Settings and alter the RGB working space popup and the color appearance changes (not the numbers) because now PS is making a new assumption about this untagged RGB mystery meat. The OFF policy and untagged docs are bad, bad, bad!
Thank you all. I agree, the "color management off" policy has always confused me. It's never really "off" photoshop is doing something with those imported color numbers. Now I know it's keeping the numbers (as they are perceived in photoshop) and assigning them to the new color space (as opposed to "converting" them).