Akie, welcome to the forum, great initial post.
Wow, wonderful pictures, especially of the cheese bread!!!!
I have no clue on type of cheese, except that somewhat similarly when my family and I lived in Syria, on road to Beirut in Lebanon there was a road stop we and locals used to stop at that sold coffee/tea and a wonderful flat bread with 2-3 similar looking different lightly melted salty white cheeses on top and served cut in diamonds on paper. It was also wonderful.
On yours sadly I do not know, was it made/sold all across or just a certain area of Japan, while hard as your childhood, to help, can you describe the flavor in any way, salty, creamy, one cheese type or mixture of two?
Thanks again for the wonderful pictures and a reminder for my memories.