1 and 2 mean that he is the only student in the school, yes.
3 and 4 can work if you change "of" to "at" or "in." I would also delete "the" from #3.
1. He is the student of our school.
2. He is the student of his school.
[Are both grammatical? Are both mean that there is only one student at school?]
3. They are the students of their school.
4. They are students of their school.
[Can we use both sentences? Does 'the students' mean 'all the students' in #3? Does 'students' in #4 mean some of the students?]
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