Learn English – Any specific word to describe a date, contains only year, month and day, but not time

When used in reference to time, date almost exclusively refers to a specific 24 hour period that begins at midnight and ends at 11:59pm. When used this way, the date will be specified with a year (CE or BCE), the name of the month, and the number of the day within that month. The day number is a positive whole number that may be used in ordinal form (first, second, third, etc.) The name of the weekday is rarely referenced when referring to a date.

When setting up an appointment, you typically use the term 'date' if the appointment is a long time away or in a formal business setting. When making appointments with friends, especially if they are happening within one week's time, you would use the term 'day'.

You ask a friend "What day do you want to have lunch?" The receptionist at your doctor's office will ask you "What date would you like to have your next appointment?"

You can say that your shift spans two days.

span verb 2 Extend across (a period of time or a range of subjects) ‘A complete planning cycle should ideally span a period of about three to five years.’ - ODO

Here's a usage example (emphasis, mine):

We are pleased to be supporting this Bill, which amends the Holidays Act 2003 to ensure that when a work shift spans two days, at least one of which is a public holiday, an employer and employee can enter into an agreement to transfer the public holiday to cover one whole shift. - Flavell: Transfer of Public Holidays Bill, Scoop