There is a big difference between saying 2 people have shared values, and saying that 2 people have similar political views, goals in life, etc.
"They both want a legal path to citizenship for immigrants"
Sure, but one group only wants that for a small number of specific types of immigrants (e.g. high "value" immigrants with specialized skills), and another group wants that for basically anyone who isn't a rapist (I am in the second group, to be clear).
"They both want their kids to have good opportunities in life"
But one sees that as a zero-sum proposition, where someone else's kid getting a good job means their kids can't get that job, and see that as a normal part of life, and another group will see it as a failure to actually provide on the promise of that opportunity.
"A majority of people acknowledge anthropocentric climate change"
But one group is unwilling to change legislation in ways that will overtly impact their lifestyles in order to counter it, and another wants massive legislative updates in order to rein in the (lifestyle) companies most responsible.
If I had to put a label to the problem, if say it's an intrinsic issue of individualism versus collectivism.
If you abstract end goals/ values high enough, everyone wants the same thing, but that is hugely deceptive to the reality of our divisions, which are about what those values should look like in implementation.