raising a family remains a key objective

The second version is correct.
The first version requires "which is the same result" before "as that" in order for it to make sense. In the second version the "it" refers to "the objective" and so it already makes sense.

I would say for those in other cultures instead of of other cultures.

as it is of other cultures ...I get lost with this. Is not it should be "for" instead of "of" ? what the native speakers say ?

No, that is correct, and logical in English. Let's examine the sentence again:

Research shows that marrying and raising a family remains a key objective of most young Americans, as it is of other cultures.

And now let us write it out in more detail so that you can see the logic being applied here:

Research shows that marrying and raising a family remains a key objective of most young Americans, as it is a key objective of other cultures.

The first example simply eliminates the need to write out "a key objective" twice.

Yea. Looks better now. Thanks for your time.