How to tag the side of the cycleway/footway on a segregated path?

I don’t see why not. The issue is, though, as @Minh_Nguyen pointed out, this only really works properly with oneway paths, otherwise we’d need to use :forward, and :backward, or well … :both_ways. So even though there is a way to make it work with :lanes, it would not be a simple solution, so I don’t think it’s worth chasing that thought further.