Often said, yet it somehow very often excludes religion, which is not immutable, and very often includes simple matters such as intelligence, which is immutable.
I find that more often than not, when a man speaks of such “immutable" traits, what he really means is “whatever the culture I grew up in arbitrarily treats much like an ethnicity, a tribe, something to owe fealty to” — religion is that in many cultures, but intelligence is not.
And I stress how cultural it is: it is often noted that in Japan religion is not treated as such a tribe, culturally, and that Japanese persons often find it curious how residents of other cultures can owe a lifelong fealty to a single religion rather than experiencing elements of multiple religions as it be suitable.
One could argue that such things are so associated with immutability merely because they are treated as a fealty in their respective cultures, a lifelong allegiance, and that switching sides is considered such a betrayal, that they are culturally made immutable, even though there is obviously no reason why religion or political affiliation should be.