No, “what’s the date today” and “what day is it today” are not interchangeable. These are two different questions, and will yield two different answers.
When asking for the date, what you are actually asking for is the calendar date. If you ask somebody for the date, you will get “March 23,” or “June 11,” or maybe something exciting like “December 25.” If you want the day of the year, this is the question you need to ask.
Alternatively, if you ask for the day, what you will generally get is the day of the week. If want to know if it’s Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday, this is what you should ask. Depending on the context, a person asking for the day might also be asking if it is a specific day. For example, if it is obviously Halloween and there are pumpkins and spooky decorations set up in every yard, a person unfamiliar with Halloween might ask, “what day is it?” Overall, however, the former connotation is more common.
This is a very subtle difference, but an important one. Even people who are fluent in English and have spoken it their entire lives often mess this up, especially since the two words sound so similar. If you ask either question, people will often ask to clarify whether you mean “day or “date.” There is a significant difference between the two, so make sure you get the right one.