Matt, I think what you're seeing is that you've constrained two points together. That constrains the part from translation, but does not address rotation. What you might want to do is add the mate to the origin, then add a fix mate, then delete the coincident mate so it's not overconstrained. That should stop all movement of the part.
Your explanation makes some sense. But, as you can see in the assembly, if I make the mate when not in "Multiple Mate" mode, ie, single mates, and mate part origins to the assembly origin they are fully constrained with 1 mate.
If you look at the part mates in the assembly, you will see that the top part has 1 mate, part origin to assembly origin, but the part is free to move because I created that mate in "Multiple Mate Mode". However the two other parts in the assembly only have 1 mate, part origin to assembly origin, and are fully constrained. If someone sent me that assembly without some sort of explanation, I'd be very confused about why the mates are behaving differently. I now know how it works, but I'm asking if this is intended behavior, and if so, why? Or is this some sort of bug? I do a lot of part origin mates to the assembly origin... And when I say a lot I mean every part in my assembly is mated to the origin with thousands of parts... So, having Multiple Mate mode work "correctly" would be a big time saver for me.
Matt Peneguy when you do a single origin to origin mate there is an additional tick box that appears called "Align Axes". When this is ticked there is actually more than one mate added in the background hence the locked position.
When you use multiple mate mode this tick box does not appear (or rather it disappears when you select more than one entity), so the additional 'in the background' mates are not created, leading to parts that still rotate as Walter describes.
I believe this is an intended limitation however I think you have a good enhancement request!