What exactly does "essence" mean?
The Biblical significance of essence is described in experiential terms. Here is one example:
John 14:16-17 NET
Then I will ask the Father, and he will give you another Advocate to be with you forever— the Spirit of truth...
Receiving the Spirit of Truth (1) comes after Jesus (2) asks the Father (3). From the perspective of truth, those who receive the Spirit of Truth do so only because Jesus, who is the Truth, (14:6) asks the Father. There is no direct mention of truth and the Father, yet Jesus says He and the Father are one. Therefore, Jesus who is the Truth, asks the Father, who likewise is the Truth, to send the Spirit of Truth. To say truth is an essence would fail to capture the dynamic component. Each of the three has a different part of the believer's experience of truth, and yet truth is not only singular; it is the same for all believers.
There is an additional component to the Spirit of Truth whom Jesus calls, allos (another) parakletos. Another parakletos means a second one. The first is Jesus:
1 John 2:1 ESV
My little children, I am writing these things to you so that you may not sin. But if anyone does sin, we have an advocate with the Father, Jesus Christ the righteous.
Greek has two words which mean "another, allos and eteros. Allos is another of the same type; eteros is another of a different type. Even though Jesus is in human form, He calls the Spirit allos not eteros. Despite what could be seen as different, man and spirit; Jesus says they are the same, despite differences in appearance. The meaning of the Greek is straightforward and unambiguous. The difficulty in defining essence comes when the definition is divorced from experience and given in philosophical terms.
What is the significance of sharing one divine essence/nature?
If humans have a common essence/nature but are not in perfect union, why does sharing one divine essence/nature mean the three persons of the Trinity are in perfect union?
Since man is marred by sin, it is usually not a fruitful approach to use the human experience to understand the divine essence/nature. However, the OP has placed the issue in a contrast of imperfection (man) with perfection (God).
Man is not only imperfect; there is that which is distinctly different, male and female. A perfect union is one in which that which is distinctly different acts as one. When that occurs, as in creation or love (see below) the result is the evidence of perfect union. The penultimate evidence of sharing one divine nature is making children of God. Father, Son, and Spirit are distinctly different, and each acts in distinctly different ways to give authority to those who believe in becoming children of God. If any one of the three was not in perfect union with the others, there would not be children born of the will of God.
Here, as before, it is the experience from the union which both demonstrates and defines the divine union.
What is it that makes the perfect union (and the biblical basis for it)?
Two things stand out which show the perfect union of the Trinity.
One may be found in the creation of man:
26 Then God said, “Let us make man in our image, after our likeness... 27 So God created man in his own image, in the image of God he created him; male and female he created them.
The plan (1:26) is described using plural pronouns, us, our. The work (1:27) is described as singular. Also, the planning speaks of man, אדם, literally Adam and the work speaks of man, האדם, literally the Adam despite the physical reality there are male and female. The exact meaning of being made in the image of God is debated. Yet, God can speak of creating the Adam as if there is a perfect union, male and female. That is, the Adam is written as one despite the fact there is a male and a female. The image of God is one of union between two which are distinct yet are one.
Another may be found in what could be considered a definition of the essence of God:
1 John 4:7-8
7 Beloved, let us love one another, for love is from God, and whoever loves has been born of God and knows God. 8 Anyone who does not love does not know God, because God is love.
God is love and this singular love is given by the Holy Spirit and preserved by Christ:
and hope does not put us to shame, because God's love has been poured into our hearts through the Holy Spirit who has been given to us.
Romans 8:35, 39
35 Who shall separate us from the love of Christ?... 39 nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord.
One can look back on Genesis and say the planning of our, God's image and our, God's likeness is accomplished through the perfect union of love from God poured into the heart by the Holy Spirit and preserved in eternity by Christ Jesus our Lord.