I did some research. It's real. It's apart of a recent line of Tamagotchi's released called "Tamagotchi School" or "TamSuku". Tamagotchi School begins with you choosing your first student and what you want to teach. You can teach Japanese Language Studies, Math or Science. While you choose your first student you actually start out with a total of six students in your classroom. Each of your students have a main subject they study, regardless of what you teach. Your main environment is in the classroom with a chalkboard in the distance. Students pass in front of the chalkboard all day long, many times one student will stick around for several minutes.
As your teaching increases, so does your classroom size. New students come by and ask to join you a few times a day. Try to make your classroom as large as possible by being a good teacher and helping your class get high marks.
Basically it's a school simulator. Sounds boring to me.
It's real. It's a Japanese model Tamagotchi called the TamaSuKu (TamaSchool) which came out on November 23, 2006.
The TamaSchool is not a virtual pet like previous Tamagotchis. You don't raise a creature from an egg to adult-hood. Instead, you are the teacher, and your success is measured by how well you keep your class on track. Your class size grows as you become a good and popular teacher.
Case artwork is shown here: click here The pattern of the one in the auction link you provided is shown on that page. The pattern is called "School Check Pink".
Being a Japanese model, the instruction manual, and all text on the toy itself will be in Japanese. If you can't read Japanese, you can probably still enjoy playing with it, if you use the excellent English language guide here: https://www.mimitchi.com/tamasuku/index.shtml