I've created a POPnet that spawns 1 particle randomly on a surface, and used a copy sop to duplicate that POPnet. In the POPnet's random seed value I've copy stamped in the $CY value. It seems to work great on the first frame, making copies of the particle system each with one particle in a random spot. But on frame 2 the particle from each system snaps to the same spot.
I would expect the particles to just stay at their spawn location.

Is there any reason the particles are snapping to one spot?

particles need to be simulated
it means that each frame depends on the previous
and when the POPnet reaches for the information from the previous substep it reads only the last cooked version because it didn't cache all the copies
i can't imagine how it could keep track of every stamped version and simulate it differently

is there any reason why not use more than one point in POP simulation?

Ok, that makes sense.

I was hoping to use each system to make a separate trail. Not the most efficient way to go about it, but I wanted to see if it could be done that way.

So would the lesson here be:
Don't try to copy stamp values into a POPnet?

Another interesting thing is, I would expect the ID on each particle to be 0, but they're sequential to the number of copies…

frogspasm_deux
Ok, that makes sense.
…
Another interesting thing is, I would expect the ID on each particle to be 0, but they're sequential to the number of copies…

your expectations are correct (particle IDs start at 1, so every particle should have ID = 1)
and it has
do not confuse point numbers and IDs

in the viewport you have checked display point numbers, point numbers are unique for each point/particle
but all your particles have attribute called ID and it is the same due to copying
you can open spreadsheet for your copyOp to see it
or you can hit D in your viewport for options
create a custom attribute text for id to display ID number next to each particle in viewport

Thanks for the explanation and tip!