No, the purpose of this beautiful poem is not to promote pornographic passions or to incite to lust and immorality. Here is part of a comment in the introduction to The Song of Songs in my 2000 edition of the New International Version Study Bible:
To find the key for unlocking the Song, interpreters have looked to prophetic, wisdom and apocalyptic passages of Scripture, as well as to ancient Egyptian and Babylonian love songs, traditional Semitic wedding songs... the Song belongs to biblical wisdom literature and that it is wisdom’s description of an amorous relationship. The Bible speaks of both wisdom and love as gifts of God, to be received with gratitude and celebration ... Rather, it views the Song as a linked chain of lyrics depicting love in all its spontaneity, beauty, power and exclusiveness – experienced in its varied moments of separation and intimacy, anguish and ecstasy, tension and contentment.
The Song of Songs (or Song of Solomon) extols the virtues of love between a husband and his wife. The poem is a dialogue between a husband (the king) and his wife (the Shulamite). It starts with the courtship, goes on to the wedding and then concludes with the relationship between husband and wife with the passing of time.
I found an article that explores the meaning and symbolism within this beautiful love story, which concludes thus:
As the song ends, both the husband and wife are confident and secure in their love, they sing of the lasting nature of true love, and they yearn to be in each other’s presence. Source: https://www.gotquestions.org/Song-of-Solomon.html
EDIT: Some scholars have suggested that the Shulammite is Abishag: “Return, return, O Shulammite, that we may look upon you” (Song of Solomon 6:13). This is because of a similarity between the words Shulammite and Shunammite: “So they sought for a beautiful young woman throughout all the territory of Israel, and found Abishag the Shunammite, and brought her to the king” (1 Kings 1:3). My NIV Study Bible notes that in ancient Semitic languages the letters ‘l’ and ‘n’ were sometimes interchanged. https://www.gotquestions.org/Solomon-Adonijah-Abishag.html