Hi @andsoforth,
As far as I know, an individual assignment or quiz item can only be put into one module at a time, as you have discovered. If the modules are weekly/dated type things, you could move it from this week's module to next week's manually when the next module started, but that's about the bets idea I can come up with to do something close to what you'd like.
The way modules work in Canvas, having the same assignment/quiz in multiple modules would present some challenges. One I can think of off the top of my head would be that the previous/next buttons, as currently done, wouldn't know where to send students. Another complication could module requirements/completion configuration. Jist bringing these things up because it might seem this would be an easy thing for Instructure to add/fix, but the behind-the-scenes reality is probably much more complicated.
Hope this does help a bit at least though.
-Chris