How common is it to fast for 40 or more days among Christians?
No single denomination practices Lent as the OP asks about: absolutely no food for 40 days. But many denominations do practice fasts for 40 days, although they differ one what the fast is from. It is far from 100%, but is nevertheless quite commonplace!
The reason for a Lenten fast, is to prepare oneself in mind and body for the glorious Feast of the Resurrection of Our Lord Jesus Christ on Easter Sunday!
The Orthodox fast traditionally all Lent.
Eastern Orthodoxy and Byzantine Rite
In the Byzantine Rite, i.e., the Eastern Orthodox Great Lent (Greek: Μεγάλη Τεσσαρακοστή or Μεγάλη Νηστεία, meaning "Great 40 Days" and "Great Fast" respectively) is the most important fasting season in the church year.
The 40 days of Great Lent includes Sundays, and begins on Clean Monday and are immediately followed by what are considered distinct periods of fasting, Lazarus Saturday and Palm Sunday, which in turn are followed straightway by Holy Week. Great Lent is broken only after the Paschal (Easter) Divine Liturgy.
The Eastern Orthodox Church maintains the traditional Church's teaching on fasting. The rules for lenten fasting are the monastic rules. Fasting in the Orthodox Church is more than simply abstaining from certain foods. During the Great Lent Orthodox Faithful intensify their prayers and spiritual exercises, go to church services more often, study the Scriptures and the works of the Church Fathers in depth, limit their entertainment and spendings and focus on charity and good works.
Among the Oriental Orthodox, there are various local traditions regarding Lent. Those using the Alexandrian Rite, i.e., the Coptic Orthodox, Coptic Catholic, Ethiopian Orthodox, Ethiopian Catholic, Eritrean Orthodox, and Eritrean Catholic Churches, observe eight weeks of Lent.
In Ethiopian Orthodoxy, fasting (tsome) lasts for 55 continuous days before Easter (Fasika), although the fast is divided into three separate periods: Tsome Hirkal, eight days commemorating an early Christian figure; Tsome Arba, 40 days of Lent; and Tsome Himamat, seven days commemorating Holy Week. Fasting involves abstention from animal products (meat, dairy, and eggs), and refraining from eating or drinking before 3:00 pm. Ethiopian devotees may also abstain from sexual activity and the consumption of alcohol.
As in the Eastern Orthodox Churches, the date of Easter is reckoned according to the Julian Calendar, and usually occurs later than Easter according to Gregorian Calendar used by Catholic and Protestant Churches. - Lent
Although the Catholic Church of the Latin Rite has reduced the Lenten fasting obligation to two days a years, it remains a staple amongst Roman Catholics that hold onto the traditional norms, especially those attached to the Mass of Pope St. Pius V.
The Syriac Church still observes the Fast of Nineveh.
The Daniel Fast is a religious partial fast that is popular among Evangelical Protestants in North America in which meat, wine, and other rich foods are avoided in favor of vegetables and water for typically three weeks in order to be more sensitive to God (the way he speaks to us and moves in our lives).
The Black Fast is a severe form of Christian fasting, especially amongst Catholic exorcists. It is the most rigorous in the history of Church legislation and is marked by austerity regarding the quantity and quality of food permitted on fasting days as well as the time when such food is legitimately taken.
The following may be of some interest: