I believe that that setting only affects the output to the display. It doesn't affect the document setting, and therefore has no effect on printing. If you want to print a rich black, you must define the rich black in the document.
Which document color mode ist it?
On what kind of device do you print?
Does it have its own color adjustment settings?
Rich black is only for display monitor
can you provide more details of how you are printing?
have you tried saving it as pdf and printing it from the pdf viewer? will it come out the same?
If you're printing within AI i would bet you have something changed inthe color management section which sets the printing profile settings.
Check what settings you have and see if you have multiple printer profiles first
What is the color breakdown for your black color?
The Black color that are you printing is a Rich Black?
Yes - is the OP using 100% straight black or is it a mix of CMYK, RGB etc.
If the OP wants the color to come out as "rich black" then change the swatch to a true rich black CMYK like
C 60, M 40, Y 40, K 100
Rich black is not only for display monitors.
The Appearance of Black settings (Printing/Exporting options) does apply to printed or PDF documents IF they are RGB.
If the OP is in a CMYK color space, that preference will do nothing for them.