The short answer is…no. A student doesn’t have to know their right hand vs their left hand in order to learn to play the piano.
As a piano teacher you will teach the student which hand goes to which staff, which notes, and (generally speaking) which side of the keyboard each hand will use. Whether a student identifies their hands with the titles of “left” or “right” is essentially irrelevant.
In referring to the student’s hands you can refer to their “dominant” and “non-dominant” hand, or to the “bass hand” or “treble hand,” or to the “down hand” or “up hand.” Or any other terminology which works for you and your student. There are many different terms or ways in which to refer to the hands when teaching your student.
Another option is to teach the concept of “right” and “left” hands along with your lessons about hands, staves and notes. And then a student can learn all of it together at once. Win-win-win!
