What is the ruling on someone who inadvertently broke their fast one day in Ramadan, then made it up and inadvertently broke their fast again on that makeup day due to vomiting, and continued to break their fast, thinking it was like a supererogatory fast?

Since breaking the fast occurred by mistake, there is no sin upon you, for the Prophet, peace and blessings be upon him, said: "Indeed, Allah has pardoned for my Ummah their mistakes, their forgetfulness, and what they are forced to do." Nor is anything required of you for continuing to break the fast on the day of making up (a missed fast), because no expiation is entailed for breaking a make-up fast, even if it was done intentionally.