There are a couple of scriptures that indicate that this world is unique among worlds. Ironically, one of the ways this world is unique is that it is not unique, as we read in D&C 76.
D&C 76:23 For we saw him, even on the right hand of God; and we heard the voice bearing record that he is the Only Begotten of the Father—
24 That by him, and through him, and of him, the worlds are and were created, and the inhabitants thereof are begotten sons and daughters unto God.
Joseph Smith rendered this revelation into poem form, and he rendered these verses thus:
And I heard a great voice bearing record from heav’n,
He’s the Saviour and only begotten of God;
By him, of him, and through him, the worlds were all made,
Even all that career in the heavens so broad.
Whose inhabitants, too, from the first to the last,
Are sav’d by the very same Saviour of ours;
And, of course, are begotten God’s daughters and sons
By the very same truths and the very same powers.
Thus the inhabitants of all worlds are saved by the same Savior as ours. In that sense, we are not unique at all.
But that makes us unique at the same time, for this is the world, among all others, on which Christ was born, lived as a mortal, performed the atonement, died and was resurrected.
Certainly all other worlds can learn of those events just as the Nephites did. And certainly he visits them when they are worthy just as he visited the Nephites.
This world was also the mortal home of Michael, who the scriptures call the archangel, and who drove Lucifer out of heaven by the power of the lamb. Whether there are other archangels at the head of other worlds, or whether Michael was unique in that way, we do not know.
But there is one other way this world is certainly peculiar, for Christ said to Enoch:
Moses 7:35 Behold, I am God; Man of Holiness is my name; Man of Counsel is my name; and Endless and Eternal is my name, also.
36 Wherefore, I can stretch forth mine hands and hold all the creations which I have made; and mine eye can pierce them also, and among all the workmanship of mine hands there has not been so great wickedness as among thy brethren.
Given what we know, it is unlikely that day was actually the most wicked day this earth has, or will, endure.
Thus this world seems to be uniquely and distinctly wicked among the Father’s creations.