Since you don't find the concept of annulment in the Bible, it is not a term you normally hear used by members of the church of Christ. Divorce is a sin, and under the new covenant, Christ said it is only permitted in God's eyes in the event of adultry. The most conservative view is that if a person divorces for other than adultry, they should remain single. If they remarry other than their first spouse, they cause their subsequent spouse to commit adultry because that is what the Bible says.
Since each congregation of the church of Christ is autonomous, some are more tolerant of divorce and remarriage for reasons other than adultry, and some define adultry in a broad sense such as when a spouse is no longer faithful based on either thoughts or deeds.
The concept of annulment is found within marriage itself, because a valid marriage requires the free consent of both parties. If one of the parties cannot consent because that party is mentally impaired or forced at gunpoint (to give two extreme examples), that marriage, although facially valid, would be subject to annulment because of a failure of consent. It is not divorce, but simply a recognition that the element of consent was lacking, and an acknowledgment that a true marriage did not take place.
I have no idea whether the Church of Christ, or some individual Church, conflates the concepts of divorce and annulment. I do not even contend that there was a failure of consent in Fred Thompson’s first marriage that would subject it to annulment. I do not know. I only observe that, because of their very young age and the pregnancy, this marriage might well be one in which the requisite consent was lacking. If that were the case, the marriage would be invalid, and Fred Thompson would be considered by many churches to be in his first marriage.