In talking with friends, it would seem like our experience was extra crappy and yours is uncharacteristically mild. So the reality may be that many have to wake up at least once in the night to feed, so 2nd sleep kinda makes sense to me here. Not that I'm saying it's where the phrase comes from.
I've often thought that much of the difference between poor/adequate/great parents is the personality of the kid(s). Have a really easy one and then you think you're some of rockstar parent. Then number two comes along and shatters your delusions.
To be honest, with my two kids, how much they eat influences their sleep heavily as well. Unfortunately, my first has always eaten too little (he'd happily have his broccoli too, but never enough to be consider him fed), and the second is very picky (she eats a lot of what she likes, none of what she doesn't). When they eat well, they usually sleep well too.
Turns out if you stop feeding them or going to them in the middle of the night, they adapt and sleep through in less than a week.
I am sure we could train them with some sort of food scarcity approach (this is the only thing there is to eat; now is the only time you can eat) to teach them to eat enough of the food that's there, but that'd take a psychological toll for a few weeks on us that we aren't able/willing to take on.
There's a reason parents pick up this version (i.e. the "let them cry it out" version, not the "create a safe and comfortable environment to allow them to self-regulate when they wake up" one) of so-called sleep training from books, "experts" and now apparently also training centers, whereas co-sleeping needs to be actively discouraged to stop parents from doing it intuitively.
I'm not saying every parent who doesn't co-sleep with their child is engaging in child abuse, but many mainstream forms of "sleep training" (especially the informal ones) very much boil down to "neglect your infant until they learn not to broadcast their needs because nobody will take care of them".
I'm also not saying that OP's account is representative of all co-sleeping parents. Co-sleeping (with breast-feeding) simply allows for reducing interruptions from nightly feeding in a way that is hardly replicable without it.
My wife is a complete zombie, I don't think she had more than continious 3 hours of sleep for at least 6 months now. I would cut my own arm off to make this situation improve at this point.
Hang in there! It won't be that way forever.
Sometimes that's not an option, but this worked great with our first (we couldn't do it with our second, so my wife is more of a zombie this time even though the daughter is a better sleeper on average).