I immediately got nerd-sniped by this...

A heuristic could try all kinds of fun force-directed placement stuff. For an exact solver, I think your best bet would (unfortunately) be to finely quantise the possible positions and orientations of each piece, describe each piece by 2 sets of positioned unit vectors (the vectors in one set start at the centre positions of each "hole" and are directed towards the centre of the piece, the other set is similar but holds the "plug" positions and orientations), and say that a partial solution is feasible iff every "plug" position is sufficiently far from all other plugs, and either sufficiently far from all other holes or within some small tolerances (in terms of both distance and angle) of some other hole. Actually, you would probably also need to treat the piece body as a "nothing else can occupy these positions" constraint. I say "unfortunate" because the position and orientation grids would need to be very fine relative to the size of the pieces, leading to a large state space thus long running times.

I think it would be fun to look for solutions that maximise the "badness" (e.g., sum of distances between matched holes and plugs) :)