What we must 1st understand about Satan is what Jesus says about him,
"He was a murderer from the beginning, and abode not in the truth,
because there is no truth in him. When he speaketh a lie, he speaketh
of his own: for he is a liar, and the father of it.(John 8:44)
Therefore, we must not trust what comes out of a liar's mouth.
Since Jesus doesn't take issue with the devil's statement, we can conclude that the devil certainly 'exercised' authority over the kingdoms of the world, yet there is nowhere in Scripture that records the devil being given authority by God; that authority was given to Adam,(Gen. 1:28)
"And God blessed them, and God said unto them, Be fruitful, and
multiply, and replenish the earth, and subdue it: and have dominion
over the fish of the sea, and over the fowl of the air, and over every
living thing that moveth upon the earth."
God gave Adam dominion, which in Hebrew : רָדָה(radah) along with : כָּבַשׁ(kabash) means to "reign over and bring into submission". Therefore, man rightly exercising his authority before God, would bring into subjection and submission all of God's creation.
When Adam disobeyed God's commandment,(Gen. 3) they "gave" their dominion to Satan; by accepting his 'offer' to eat from the Tree of the Knowledge of Good and Evil and become like him, and he, being the "strong man"(Mark 3:27), was able to bring men into "dominion"(reign over and bring into submission). God extended man mercy, prophesying that one day the woman's seed would crush the head of Satan(Gen. 3:15), but until that time worldly government remained under Satan's dominion, although Godly men from time to time have ruled.
Men indeed have subjected themselves to Satan, worshipping him in idolatry and giving themselves over to perversions and evil, in violation of God's commands. Furthermore, as Jesus addressed the Jewish leaders, those that oppose Jesus are of the devil(John 8:44) and whoever is not for Jesus is against Him(Matt. 12:30)
In Summary: Satan obtained "dominion" by taking it away from Adam & Eve. The 2nd Adam,
Christ(1 Cor. 15:47)
the second man is the
Lord from heaven.
And the 2nd Adam will reign, undoing all the work of Satan(1 Cor. 15:24-25)
"Then cometh the end, when he shall have delivered up the kingdom to
God, even the Father; when he shall have put down all rule and all
authority and power.
25 For he must reign, till he hath put all enemies under his feet"