What should one do who thought he had to make up 15 days from Ramadan 2018, and began making up 6 days, then discovered that the make-up was for Ramadan 2019 and he owed 20 days? Should he continue or restart, and is there a difference of opinion on the matter?

It is not obligatory to specify the year or the day when intending to make up for missed fasts. The intention to make up what is due is sufficient. An error in specification does not invalidate the fast. So, if one intended to make up a day from a specific year and erred, their fast is valid, and they should continue based on what they have already fasted. An exception to the obligation of specification is making up for two Ramadans, or a votive fast (fast of a vow), or an expiatory fast (fast of atonement) from different causes; in such cases, the intention to make up is sufficient without specification. It must be noted that whoever delays making up missed fasts without a valid excuse until the next Ramadan begins, owes a fidyah (expiation) of feeding one poor person for each day. An ignorant person regarding the impermissibility of delaying the make-up fast is excused and is not required to pay fidyah.