After reading the Color Management and Color Picker sections of the Blender manual, I can only assume I'm seeing a bug in Blender.
RGB values in the Color Picker node are in linear Space. RGB values in the image Sample tool are in linear space. They match: good.
Open an sRGB image in GIMP. Use the color picker tool to sample a 4-pixel-radius region that is flat and uniformly shaded. Copy the RGB data into Blender's Color Picker node. Save the render in sRGB space, open in GIMP, and sample the same region. Colors do not match: expected, since the RGB values of Blender's Color Picker node work in linear space while GIMP's work in sRGB space. Switching GIMP to linear light encoding does not seem to affect GIMP's color picker.
Repeat the above procedure for HSV rather than RGB values. GIMP's color picker displays hue in [0-360] while blender uses [0-1], so you'll have to convert. Colors match: expected, since the HSV values of Blender's Color Picker node are gamma corrected (presumably for the current display device transform).
Repeat the above procedure for hex rather than HSV values. Colors do not match: unexpected, since the hex values are gamma corrected just like HSV/HSL values. The difference is slight and I can't help but wonder if this is an aliasing error, i.e.
The actual display device is irrelevant since we are sampling data values rather than visually comparing colors.