Person B is referring to the concept of client confidentiality.
Confidential: marked by intimacy or willingness to confide
Because therapy involves discussing sensitive matters, therapists must be able to convince their clients to share information that the client may consider private.
This is made easier by the therapist's persona which is often warm, understanding, and non-judgmental. Additionally, the client knows that a therapist has a duty to not disclose information about their clients.
This exchange of information is one-sided, however. The therapist does not usually share their private thoughts with their client.
Like a therapist would be to their client, person B's mother was warm and inviting to person A and was always willing to discuss what was on person A's mind. She was attentive to person A's needs.
Also like a therapist, person B's mother did not share many details about herself with person A. There was an imbalance of information in the relationship that led to person A not knowing about person B despite person A being close to person B's mother.
Person B notes this dynamic and correctly compares it to the relationship between a therapist and their client.
Generally, you could say that person B's mother was a confidant for Person A. She was someone that person A trusted with private information, but Person A did not necessarily receive the same trust in return.