Is performing the obligatory prayer after success considered fulfilling a vow, or is it a prayer like that of other Muslims, and how are vows recognized in such a case?

Vowing to perform the obligatory prayers or other obligatory acts has no meaning, because it is tahsil حاصل (achieving what is already achieved) and committing to what is already binding. Some scholars have explicitly stated the dislike or prohibition of undertaking this type of vow. Therefore, this vow is not valid.