What is the ruling on fasting on the anniversary of a relative's death each year as a charity on their behalf, and is this considered a vow, and what are the consequences if the fast is not observed in one of the years?

Designating the anniversary of your relative's death for fasting every year is an innovation (bid'ah) for which there is no evidence. It falls under the category of additive innovations (bid'ah idafiyyah), and therefore, it must be abandoned. If it is established that it is an innovation, then fulfilling it is not obligatory if it was vowed, because a vow is only obligatory to fulfill if it pertains to an act of worship (qurbah). A vow is only binding if it is expressed in a form indicating commitment, such as saying, "For Allah is upon me such and such," and it is not binding by mere intention. The deceased benefits from the reward of supplication (du'a) and charity (sadaqah (definition)) by consensus, while scholars differ regarding the arrival of other acts.