What are the practical applications of "two will become one flesh"?

I don't agree that this is (just) an analogy.

The phrase "become one flesh" is literally true when it comes to having children, as a child inherits genetic (fleshly) material from both parents. The result of parental intimacy is, therefore, a very real fusion of their physical makeup in the child.

That said, these physical realities do point to other non-physical outcomes of the union between man and woman. These have been adequately suggested by other answers.

I would like to add that the verse is not only analogous to what it will mean for believers to be the bride of Christ (in Christ) but also to understanding the trinity. God the Father and God the Son and God the Holy Spirit are three different persons yet one God (John 10:30). Likewise, the bride of Christ are one with Him. Ephesians 5:15-32 shows a great relevance between marriage of husband and wife and how the church will be Christ's bride and is good application to your question: What does being one flesh mean?.

Marriage is a spiritual covenant made within the confines of God's will between two parties and thus doesn't require sex and should not be unholy. Being one flesh means being one in accord with the other just as Jesus came to do the Father's will. The covenant is not to be broken. This is why divorce is offensive to God. As it represents his union with the believer, what would we say if Christ suddenly changed his covenant with His body and stated that he would not come back for it as He promised? So, being one in flesh means being in unity. Each party has a different personality, each party may have a different purpose, yet, each party is bound together in unity as one.

We are blessed as blemished souls that Christ would consider us to be united with Him who is perfect.

According the Catechism of the Catholic Church:

The Lord himself shows that this signifies an unbreakable union of their two lives by recalling what the plan of the Creator had been "in the beginning": "So they are no longer two, but one flesh."

This excerpt contains a footnote pointing to Matthew 19:6:

So they are no longer two but one flesh. What therefore God has joined together, let not man separate.

Thus, according the Roman Catholic Church, the phrase "no longer two, but one flesh" refers to the indissolubility of marriage. Once God joins a man and a woman in marriage, this marriage can only be undone by the death of one of the spouses.

Obviously, there are many other truths and lessons that can be gathered from the phrase, but this is how the Catholic Church interprets the phrase's intended meaning.

In my view, this is saying that husband and wife should work together for the same goals. This goes back to the very beginning when God made Eve from Adam's flesh as a helper

Genesis 2:20-24 (NIV)

20 So the man gave names to all the livestock, the birds in the sky and all the wild animals.

But for Adam no suitable helper was found. 21 So the LORD God caused the man to fall into a deep sleep; and while he was sleeping, he took one of the man’s ribs and then closed up the place with flesh. 22 Then the LORD God made a woman from the rib he had taken out of the man, and he brought her to the man.

23 The man said, “This is now bone of my bones and flesh of my flesh; she shall be called ‘woman,’ for she was taken out of man.”

24 That is why a man leaves his father and mother and is united to his wife, and they become one flesh.

God didn't create us to labor alone.

Ecclesiastes 4:9 (NIV)

9 Two are better than one, because they have a good return for their labor:

So when we get married, we have to merge our goals and work through life together. This doesn't mean that the married couple will always see eye-to-eye. That's the beauty of God's design: we see things in different ways to accomplish the same thing.

But when we disagree on something, we don't go to our friends and complain about the other spouse. A married couple should always having each others' interests in mind. Agree on which path to take together and present a unified front to the world.

The best way I've found to agree on a chosen course is to talk it through, then pray about it with each other. Listen for God's response. Don't dismiss the other's misgivings as trivial, for it could be God's answer to your prayer (even if you don't like the answer).

And yes, I believe if you work together long enough towards the same ends, then yes, your happiness and sadness are linked (though I doubt my wife will be sad if Alabama doesn't do well this year).

It seems that "the two shall become one flesh" is regarding sex, although it could have other connotations as well.

In 1 Corinthians 6:15-17, it says

15 Do you not know that your bodies are members of Christ? Shall I then take away the members of Christ and make them members of a prostitute? May it never be! 16 Or do you not know that the one who joins himself to a prostitute is one body with her? For He says, “THE TWO SHALL BECOME ONE FLESH.” 17 But the one who joins himself to the Lord is one spirit with Him.

Here, it obviously refers to a sexual context, as it refers to joining with a prostitute. However, the fact that it refers to sex should not down-weight the significance of "becoming one flesh", but "becoming one flesh" should up-weight the significance of the sex act, considering that the term "the two shall become one flesh" is used in the context of marriage. The fact that it is also used in reference to engaging with a prostitute means that the act of sex is a very significant thing.