I have been searching and did find what looks like the same gearbox for sale. It is advertised as suitable for a 2T or 3T Celica and a T40 Corolla, but no year of manufacture stated. I have also found discussion about what to put in gearboxes, oil or ATF, but I am not clear whether this applies to my gearbox.
I have been looking for numbers on the box, to no avail. There is " 1 - 1 " stamped at the rear of the right side and what looks like a Japanese character(s) or logo further forward on the right side.