The closest thing to Jesus having a significant other is when Jesus refers to the WHOLE church as His bride. This is a spiritual marriage (or becoming one type deal).
Ephesians 5:23 "For the husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior".
2 Corinthians 11:2 "For I feel a divine jealousy for you, since I betrothed you to one husband, to present you as a pure virgin to Christ".
1 Corinthians 12:25-27 "That there may be no division in the body, but that the members may have the same care for one another. If one member suffers, all suffer together; if one member is honored, all rejoice together. Now you are the body of Christ and individually members of it".
Revelation 19:7-9 "Let us rejoice and exult and give him the glory, for the marriage of the Lamb has come, and his Bride has made herself ready; it was granted her to clothe herself with fine linen, bright and pure— for the fine linen is the righteous deeds of the saints. And the angel said to me, 'Write this: Blessed are those who are invited to the marriage supper of the Lamb'. And he said to me, 'These are the true words of God'.'"
In fact, the relationship between a man and a woman should be modelled after Christ and the church's. A man should be willing to die for his wife just as Christ died for us. The song of Solomon is meant to be metaphoric for God's love for saved mankind.
Ephesians 5:25-32 "Husbands, love your wives, as Christ loved the church and gave himself up for her, that he might sanctify her, having cleansed her by the washing of water with the word, so that he might present the church to himself in splendor, without spot or wrinkle or any such thing, that she might be holy and without blemish. In the same way husbands should love their wives as their own bodies. He who loves his wife loves himself. For no one ever hated his own flesh, but nourishes and cherishes it, just as Christ does the church, because we are members of his body. 'Therefore a man shall leave his father and mother and hold fast to his wife, and the two shall become one flesh.' This mystery is profound, and I am saying that it refers to Christ and the church".
