You ask this question if you want to know what day of the week it is - for example, Wednesday. If you ask this, you are not enquiring about the date. For the answer to be ' 12th January', the question would need to be 'What date is it today?' or 'What's the date today?' . They are different questions requesting different information.

I think since the question asks for the [day], the best answer uses a day of the week (Sunday, Monday, etc.) If you asked "What is today's [date]?", then the best answer will use the month and day number.

Answering the date instead of the day is not the most common way to answer but it is the most sophisticated answer with the most swag.