Equality is about the rights one has, and how one should be treated as an individual — equality before the law, and before society. We strive for a society where men and woman are equal, blacks and whites are equal, straights and gays are equal.
Universal equality, however, does NOT require us to believe that people are identical in personality, in intelligence, or in any of the other dimensions you've listed above. We don't even have to believe that specific groups, taken together, are identical to all other groups in mean, median, and standard deviation on these attributes.
People can be EQUAL without being THE SAME.
Edit: really bummed I missed the opportunity to make a JavaScript joke about == vs ===.