If someone asks "What day are you leaving?", would it be grammatically correct to answer with "On the 8th April." for example? Or does it have to be "On Monday."?
This is a question of what's technically correct versus what's commonly used.
What day are you leaving? Monday / next Monday / On the 8th of April.
No one really asks "On what date are you leaving?' even though it's technically correct.
What day are you leaving? Answer should be a day of the week
What date are you leaving on? The 8th of April.
Often, when people ask the day, it's the day of the week (Monday). When they ask for the date they are looking for the actual date, April 8th, in your example. A simple way to avoid confusion is to give both answers, "Monday, April 8th".