Is maintaining the remembrance of "Subhan al-Kareem" (Glory be to the Most Generous) after vowing it to Allah considered an innovation (bid'ah) since it was not reported from the Messenger of Allah, peace and blessings be upon him, and what are the consequences for the one who made the vow if that is the case?

A man's mere recommendation of a specific dhikr (remembrance of Allah) does not obligate you to adhere to it, for what is obligatory is what Allah has made obligatory. Fulfilling a promise is recommended, not obligatory, especially if no harm results from its ترك (abandonment). As for the saying "Subhan al-Kareem" (Glory be to the Most Generous), it is a lawful dhikr because "al-Kareem" is one of Allah's names. And whoever covenants with Allah to perform an act of obedience, his covenant becomes a vow and fulfilling it becomes obligatory. The fact that the number is unspecified (not less than 500) does not make it an innovation.