All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Addressing a person by saying 'my dad' or 'my father' is permitted provided it is only for honoring him, showing affection to him, or respecting him. But if one uses this expression to deny his original lineage, then it becomes forbidden.
Ibn Abbas narrated: “The Prophet sent ahead some boys from Banu Abdul Muttalib on donkeys on the night of al-Muzdalifah. He began to pat our thighs (out of love) and said: “O my sons! Do not throw pebbles at the Jamarah till the sun rises.” [Ahmad, Abu Daawood, and An-Nasaa-I]
Shaykh Al-Albaani graded this Hadeeth as Saheeh (authentic).
It is also narrated in Saheeh Muslim from Anas that the Prophet called him by saying; “O my son.”
Therefore, the threat in the Hadeeth mentioned in your question applies to the one who denies his original lineage.
As for your last question you may inform the people that this man is your stepfather. You should even bring to the knowledge of people that he is not your original father.
Allah knows best.