Here are some ways of thinking about the sentence "No me está entendiendo mi punto":
Note that one can say in English "You're not understanding the point," or "You're not understanding my point." Just as "me" is used for body parts, Advisor A could have said, "No me está entendiendo el punto," and the indirect object would show who is affected by the lack of understanding: me. Having both the possessive and the indirect object present provides some added emphasis.
Here, punto is what Pablo called an inalienable possession in the sympathetic dative, in his canonical answer.
That optional indirect object is also functioning as an ethical dative. Pablo wrote:
The ethical dative is a dative that shows a certain interest, concern or involvement on the part of the referent. It sometimes overlaps with the sympathetic dative.
What type of involvement does the speaker have in this case? How does he feel about the other person not getting the point? He feels frustrated. The feeling that the speaker has, about the utterance, comes out through the use of the indirect pronoun.
I don't know what your first language is, but in case this helps: in English, a speaker can attach feelings to a statement in an analogous but grammatically somewhat different way, e.g. "You always go and distort things on me," or "She up and died on me."
Let's talk about body parts for a moment. I don't know a rationale for this, but the fact is that body parts and possessive pronouns don't go together in Spanish. We don't say to a small child before crossing the street, "Agarra mi mano." We say, "Agárrame la mano," which means Hold my hand.
There are several well known songs whose main line is Ponme la mano aquí. (For an explanation of the lyrics of what appears to have been the first such song, see http://memoriaflamenca.blogspot.com/2011/09/bulerias-de-la-paquera.html:
[...] un poema del poeta asturiano Alfonso Camín en que el estribillo dice: "ponme la mano aquí Macorina", que es un estribillo muy antiguo cubano de una rumba-son en el que un herido de guerra le pide a Macorina que le ponga la mano ahí donde le duele.
(Macorina was a real person in Cuba, of legendary beauty.)
Songs with the line "Ponme la mano aquí" (Put your hand here) abound, with many different styles and interpreters. If you listen to several recordings I think this will help ingrain in you the instinct to use an impersonal article (lo/la) with a body part.
In addition to the songs, another way of ingraining this might be an indignant loud admonition I once heard a woman make in a crowded medium-distance bus with people crammed in the aisle. The woman's voice rang out suddenly:
¡Quítese la mano de la pierna! [Get your hand off my thigh!]
She successfully embarrassed the groper, who kept his hands to himself after that.
A somewhat similar sentence in English is "That hand better not be on that leg again on this bus."