Is my mother sinful for breaking her fast last Ramadan due to bleeding that was initially of an unidentifiable nature, then further bleeding occurred which resembled her menstrual period, so she broke her fast based on that? And how many days does she have to make up?

The mother must make up the five days she missed fasting. As for the day she saw a yellowish discharge before the blood, she must make it up if it was during her regular period; otherwise, she does not have to make it up. It appears that the woman is experiencing istihadah (non-menstrual vaginal bleeding) because the bleeding exceeded fifteen days, and there was no minimum period of purity between the two bleedings. A woman experiencing istihadah should refer to her regular habit. If she does not have a habit and can distinguish menstrual blood, she should count the days she distinguished as such. Otherwise, she should estimate six or seven days. She must make up the prayers she missed during the days when the blood was deemed istihadah.