Is my mother's continuation of prayers, after making up all the missed prayers that she vowed to God to make up as long as she lives, valid, and is that considered a vow?

The majority of jurists hold that whoever intentionally abandons prayer after reaching puberty is obliged to make up for it. If the number of prayers is unknown, one should pray what one believes to be the most probable number. Prayers missed before puberty do not require a make-up. Some scholars, however, are of the opinion that one who intentionally abandons prayer is not obliged to make up for it but rather to repent. Your mother should frequently seek forgiveness and perform supererogatory prayers, in accordance with the Almighty's saying: (And indeed, I am the All-Forgiving to whoever repents and believes and does righteousness and then is guided) Taha/82. As for her vow to make up for missed prayers, if she has made them up, she has fulfilled her vow and is not obliged to continue, as a prayer is made up only once. Her phrase "as long as I am alive" means that she will fulfill her vow as long as she is alive without interruption. If she wishes to continue praying with the intention of supererogatory prayers, this is a good deed, for the Prophet, peace and blessings be upon him, said: "Prayer is the best subject, so whoever can do much of it, let him do so."