Only by putting a rectangle of the desired color on the bottom layer and sending it to the back. InDesign does have a paper color for its artboards.
I'm not sure what "unique effect" you're talking about, but PageMaker's Paper color was functionally the same thing as White in Illustrator. Naming the default white color "Paper" arguably makes sense in any print-centric program, because in an ink-and-paper world (i.e.; a subtractive color model), it really means "no ink."
But it's not a big deal, and nowadays most historically print-centric programs are being multi-purposed to include web-centric work. Just imagine the garment-rending outcry from web-only users if white suddenly became labeled "Paper." ;-)
What's really needed in this regard (and has been universally missing across the entire gamut of physical-media graphics programs) is the awareness that in the real world, ink and other subtractive color media are just as often opaque as translucent. Now that would be a significant and long overdue advancement!