Adhikāri means the one who is qualified. On hearing this, there is ākāṅṣā, expectation, in our minds, as to qualified for what?
So, adhikāri is a contextual word. This has to be understood.
Adhikāri in this context means, one who is qualified for ātma-vidyā, self-knowledge. And a number of the qualifications are listed.
A key point to note is, it should not be given to one who is not an adhikāri. Kṛṣṇa says in Bhagavad-gītā 3.29, that one who is not discriminative, one of the qualifications for being an adhikāri for ātma-vidyā, should not be distrubed.. he should be left alone.
Ātma-vidyā is to be given only to those who seek it, and amongst those who seek it, to only those who are qualified.