What is the cultural mandate? Who is it given to?

The cultural mandate is the command to exercise dominion over the earth, subdue it, and develop its latent potential (Gen. 1:26-28; cf. Gen 2:15). God calls all humans, as those made in his image, to fill the earth with his glory through creating what we commonly call culture.
The cultural mandate is given to all people. In Genesis 1:26-28, it’s given to Adam and Eve as the only people and as representatives of all humanity. In Genesis 9:1 it’s given to Noah as the representative of all humanity.
Therefore, the cultural mandate is not just for the people of God. Moreover, it’s not uniquely tied to the gospel or the great commission (Matt. 28:18-20), which is a distinct mandate given to the people of God alone.
Having said that, only the true people of God (the church) will be able to fulfill the cultural mandate as it was intended—according to the desire to give glory to God.