I think the title is essentially my question. Fast days are the only time when a haftorah is said during mincha. What makes fast days unique to have a haftorah recited then?
Tosfos to Megila 21a “Hakorei” explains why the haftorah is read at Mincha.
והטעם שמפטיר במנחה בתענית ולא בשחרית משום דכתיב בה שמרו משפט ועשו צדקה (ישעיה נו)ואגרא דתעניתא צדקתא לעת ערב ומש"ה נכון לאומרו בערב אחר שעשו צדקה
He says that the reward of the fast-day is the charity given on that day. It could be assumed that by the time of mincha, people would have given their charity and so it would be appropriate to read the haftorah (Isaiah 55 & 56) which urges us to שִׁמְרוּ מִשְׁפָּט וַעֲשׂוּ צְדָקָה literally “Keep justice and practice righteousness” but צְדָקָה is also understood as tzedaka (charity).
(FYI see See Rambam Matnot Aniyim - Chapter 9: Halocho (4) :On fast days, we distribute food to the poor. (Commentary: On fast days, it was customary to distribute food to the poor after the evening service at the conclusion of the fast. Since the poor would look forward to this meal and rely on it to break their fast. If it was not provided to them, they would go to bed without food (Rashi, Sanhedrin 35a). ))
From here one could deduce that the reason to read this haftorah is to emphasize the meaning of the fast day. Relevant extracts from the haftorah to justify my assertion:
Seek the Lord when He is found, call Him when He is near.
The wicked shall give up his way, and the man of iniquity his thoughts, and he shall return to the Lord, Who shall have mercy upon him, and to our God, for He will freely pardon.
So says the Lord, Keep justice and give charity, for My salvation is near to come, and My benevolence to be revealed.
The source for reading the Haftara at Mincha on a fast day is Taanis 12b end of page.
אמר אביי מצפרא עד פלגא דיומא מעיינינן במילי דמתא מכאן ואילך ריבעא דיומא קרינן בספרא ואפטרתא מכאן ואילך בעינן רחמי
ריבעא דיומא קרו ויחל משה ומפטירין דרשו ה בהמצאו
Abaye says that from morning to noon people are busy therefore the Haftara is read in the afternoon.