Well, when it says that the 'printer manages the colour' that's what it's doing.
On a Mac you get the option of the printer doing the colour matching or ColorSync colour matching.
With my printers (Canon) when the printer manages the colour the driver chooses a 'profile' dependent on the paper type selected in the driver. These are Canon's own profiles.
The only way to print without colour management is to use ACPU or, on the Mac, Preview.
If you select the application to manage the colour (e.g. Photoshop or Lightroom) then colour management by the printer is automatically turned-off. However, if you try to print a profiling target which does not have a profile attached the operating system, at least on a Mac, will impose a profile of some sort (probably sRGB).