What is the meaning of "remove" in this sentence?

On the contrary, remove as a division in school does fit here and is the intended meaning.

If, at the end of the lower fifth, the student gets his remove (i.e. passes his end of year exams), then he is removed from the lower fifth and enters the upper fifth.

In British secondary education at that time, each form (that is - year) was numbered from first to fifth. The fifth form took two years to complete, thus upper and lower fifth.
Later, an additional year was required as the school leaving age was increased. In my day, I attended lower then upper sixth.

On entering these higher years, students were allowed to drop the study of certain subjects and concentrate on others. Hence the reference to specialisation.

There is a phrasing "to get one's remove" meaning to pass or to promote a grade or exam. I can't specialise ... unless I promote.