Many if not most of these oddities about humans come down to either they don’t affect reproduction, or because humans are social animals. Menopause makes sense because a grandmother will have more successful descendants if she stops having babies and helps her daughters with the grandkids. I would guess the few other species that exhibit menopause are also social animals.
This is discussed at the bottom of the article: "Menopause: Humans, killer whales, pilot whales, beluga whales, false killer whales, narwhals one group of chimps in Uganda. No menopause: Everything else"
So, monkeys and apes (besides one species of chimps) don't get menopause, elephants don't get menopause, lions don't get menopause. If your prediction is, social animal => menopause, it's wrong 99% of the time.
It seems it might be more a combination of whether it's a social animal and how much help / how much danger the offspring is actually in. There's not a whole lot that fucks with elephants, and not a lot that successfully fucks with the young ones. And female elephants universally live in groups and collectively all protect each other. Female lions tend to do the same, and also have very few things that fuck with them.
Conversely human offspring is especially weak and incapable and susceptible to harm, and experiences a lot of threat even within the same species.
Not sure where that are of whale and chimps fit into that - flukes can also happen I guess.
A couple more points to add to this:
Most species make a baby (or babies), raise them to maturity before having another. Even a low chance of successfully raising another child is advantageous over no chance at all from turning off the baby maker. Humans, however, have children of a range of ages, making more before the first is finished. Having another inherently takes away from the ones that already exist. Having another that probably won't make it can be an evolutionary negative.
And humans have a much longer portion of their life spent on each offspring. This increases the chance of death before completion and thus makes more evolutionary pressure towards not putting effort into futile things.
Note that this works even if she is alone and doesn't help with the grandkids.