Is fasting days with the intention: "I am fasting to fulfill my duty, but if this fast is not obligatory for me, then I am fasting it as a supererogatory act," sufficient for fulfilling the obligation, without precisely specifying the number of days?

You are experiencing multiple intrusive thoughts (waswas), and the cure is to ignore them completely. Intent (niyyah) must be decisive. So, if someone fasts to make up for Ramadan with an intention that wavers between it being obligatory (wajib) and supererogatory (nafl), then their fasting does not count as make-up (qada’) and becomes a supererogatory act.

If you know the number of days you must make up, then fast all of them. If you do not know their number, then act according to your best judgment and be cautious, because best judgment is resorted to when certainty is unattainable.