I made a search on other subreddits dedicated to religion, but could not find a answer that satisfied me. As lutheranism generally advocated a christianity based on the Bibel, and since the Anointing of the Sick is present in the Bible, I was wondering which argument Luther, or any other lutheran theologian for that matter, used the exclude it as a sacrament from lutheranism?
The Lutheran view of what does and does not constitute a sacrament depends on two criteria: first, a direct institution by Christ himself with a promise of grace; second, a physical component linked with the Word of God. Therefore typically Baptism and the Lord's Supper are the only two considered to meet that definition. Sometimes Confession and Absolution is considered as a third sacrament, and in fact it is included by Melanchthon in the Apology to the Augsburg Confession (Kolb/Wengert p. 219), but it has since often been rejected or considered borderline as a sacrament by later Lutherans as it lacks the physical element such as water in Baptism or bread/wine in the Lord's Supper.
The anointing of the sick, on the other hand, has the physical component in the anointing oil, but lacks the direct command from Christ. It does have Scriptural support from James 5:14, and Luther emphasizes that only Christ and not the apostles have the authority to institute a sacrament (Luther p. 251). However, neither Luther nor the Lutheran church reject the practice. The LCMS, for example, includes a rite of anointing of the sick in its Pastoral Care Companion (p. 34). But lacking an institution by Christ they do not go so far as to consider it a sacrament.
Anointing also lacks the same promise of grace as the other three previously mentioned. Sacraments are understood to be "means of grace," ways in which the command and promise of Christ make it a sure and trustworthy place to find God's grace. While physical healing for the sick is certainly excellent and desirable, it is not on the same tier as God's grace for eternal salvation. "Hence it is useful to distinguish these rites [confirmation and extreme unction] from the previous ones [Baptism, Lord's Supper, and Absolution], which have the expressed command of God and a clear promise of grace." (Kolb/Wengert p. 220)
Commission on Worship of the Lutheran Church - Missouri Synod. Lutheran Service Book: Pastoral Care Companion. Concordia Publishing House: St. Louis, 2007.
Kolb, Robert and Timothy Wengert, eds. The Book of Concord: The Confessions of the Evangelical Lutheran Church. Fortress Press: Minneapolis, 2000.
Luther, Martin. "The Babylonian Captivity of the Church". Three Treatises. Trans. A.T.W. Steinhaeuser. Fortress Press: Philadelphia, 1960.
Schiefelbein, Kyle K. "'Receive This Oil as a Sign of Forgiveness and Healing': A Brief History of the Anointing of the Sick and Its Use in Lutheran Worship. Word and World vol. 30 no. 1, Winter 2010.