You're right that your mom knows who is on the phone. She also knows what you mean when you hold the phone out towards her. Technically, for information, you don't need to say any of the words. You could just say "phone"; or you could grunt; she would already now she is supposed to take the phone, she would already know who is on the phone.
So this is not an information question. You say the name (title) anyway, and you say a full sentence instead of a grunt. As mentioned above, this is about etiquette, and it is about expectations. Whether she listened you your conversation or not, she was not part of your conversation. A new (or at least separate) conversation starts who you speak to her (if she had been a part of that conversation, she would have heard your father say he wants to speak to her, and you would not be an intermediary). So you name the referent the first time, because that's just what we do. In many cases the grammar rules help avoid confusion, but in this case, even when confusion is not likely, we keep them out of collective habit.
In some households, of course, people will just hand someone a phone, or just scream "phone." In a different context it can have the same meaning; if you spent your entire life referring to one person as "he" in this context, it would become normalized. For example, in high school I had a friend whose brother was in prison. When the brother called, they actually did say "he's on the phone" the first time, without a referent; but "he" always specifically meant the brother when used in this context. They did not think of it as being rude to each other, therefore it was not rude.