Does the statement "I pledge to Allah not to commit what displeases Him again" upon repentance necessitate an expiation for breaking an oath (kaffarat yamin) or fulfilling a vow (nadhr) if one falls into sin again? And is expiation or a vow required for every instance this statement is uttered?

You have done well by repenting, and fulfilling a promise is among the noble virtues, while breaking it is among the vices, especially if the promise was made to Allah Almighty to abandon His disobedience. Breaking an oath made with Allah to abandon a sin is impermissible. If you commit the sin after swearing an oath to abandon it, you incur sin from two aspects: committing the forbidden act, and breaking this type of oath. In addition to the sin, you owe an expiation for breaking an oath (kaffarat yamin). What occurred is considered an oath that necessitates an expiation for each instance of breaking it, unless the matter sworn upon was a single thing and the oaths were repeated before you broke them, in which case only one expiation is required. The expiation for breaking an oath is: feeding ten needy persons, or clothing them, or freeing a slave. If you cannot find any of these, then fasting for three days.