A student of his

Of followed by possessive ordinarily means 'among'. A student of his, i.e. one among his students, is not illogical; that nose of his, i.e. one among his noses, is illogical (can you think why?) but tolerated as an idiom. You can use it.

Please disregard above post.

Illogical because it involves a a double possessive. Some writers these as questionable usage although it has a history in careful English. The Oxford English Dictionary says that this usage was "Originally partitive, but subsequently became a simple possessive or as equivalent to an appositive phrase". Fowler says "It is not easy to explain why such constructions are idiomatic: one can only assert that they are." If Fowler does not attempt an explanation, I hardly think you can expect one from me. You need not avoid using such phrases.

Your communication seems to indicate that you have more than a basic grasp of English. Why worry about ( and ask explanations of) phrases that have been termed "illogical"? Why not say "his student" and move on?

English is an idiomatic language. Idioms are perfectly correct. There is no reason not to use them.

If you are asking for English to be logical, then you are asking for trouble.