Is bathing with the intention of cleanliness considered ablution (wudu')? If not, what must one do regarding the prayers performed in that manner, and must they be repeated, given that their number is unknown?

The ghusl (ritual bath) intended for cleanliness does not suffice for wudu (ablution). Therefore, the prayers performed after such ghusl without proper wudu must be made up. If you are not certain of the exact number of prayers, continue making them up until you are reasonably sure that you have absolved yourself of the responsibility. This is the view of the majority of scholars, whereas Sheikh al-Islam Ibn Taymiyyah and those who agree with him hold that there is no obligation to make up prayers for someone who missed a condition due to ignorance or forgetfulness.