Poly-cotton shouldn't have reacted like that.

Poop, by itself, doesn't do much of anything to black dyes. At most, you might run into a spot where it weakened the cotton fibers, but it should have done that only where the feces was in direct contact with the fabric. Poo can be acidic enough to weaken some natural fibers, I've just never seen it do so after being soaked and diluted by a significant amount of water.

So, I'd expect the undies to be discolored, not something washed with them.

The only reason it matters is that if the fabric of the sock is damaged, you'll have issues getting any new dye to do much.

But that's the answer, dye. You can try washing it again to see if the color change is from residual detergent (which isn't usually going to only appear on one sock and not the things touching the sock as well), but once cotton loses pigment, you have to apply more to get it back.

Cheap option is a sharpie. The color won't match exactly, but it's cheap and fast Rit dye is the next option, but the black tends to be more of a dark gray on poly blends, in my experience. Heck, it's barely black black on cotton. And it tends to wash out to a dark gray in a few washes even then. I'm not sure where you'd get the dyes that manufacturers use, I've never had call to try. But that's the final option.

But, when you wash/rinse it to see if it's residue or whatever, cold water isn't special. Warm or hot water would dissolve the likely culprits better, but don't use a detergent. The goal is to get out any residue, not add in more soap that could be what's causing the color change to begin with.

I also noticed you said "abundant detergent". Extra laundry soap isn't beneficial. You don't really get things cleaner after a normal amount for the size of the load, you just get soap left behind.