Is it permissible to say the takbir after the prescribed prayers these days, or is it an innovation and a distraction for worshippers from the designated adhkar?

Takbir is of two types: absolute (mutlaq) and restricted (muqayyad). The absolute begins upon sighting the crescent moon of Dhul-Hijjah, while the restricted begins from the dawn of the Day of Arafah until the 'Asr prayer of the last of the Days of Tashreeq. Absolute takbir is also recited on the night of Eid al-Fitr. If your companion denies the restricted takbir at its appointed time, then he is mistaken, and the Sunnah is with the one who recites takbir. The restricted takbir does not conflict with the remembrances after prayer. However, if he denies the restricted takbir outside of its appointed time, then he is correct; for persistent recitation at that point falls under the category of Bid'ah Iḍāfiyyah (added innovation).