"What day" is the correct form when you are asking for the name of a specific day, such as "What day is it today?" or "What was the date of your birthday?" However, if you are asking about a particular time on a certain day, then "which day" would be appropriate.
What is difference between what and which?
Answer. What is used when we have unlimited possibilities as the answer. Which is used when we have limited possibilities as the answer.
Which day is today or what day is today?
What day is it?, What day is today? or What day are we in? would be correct.
How to ask what day is today?
How to ask the date. If you want to ask what the date is, you can say: What's the date today? or What's today's date? What's the date tomorrow? or What's tomorrow's date?