We have excellent contemporary records from the period of Akbar. We can look upon Akbarnama and Ain I Akbari as official court histories of Akbar. The works of Badayuni and Sirhindi were contemporary narratives that were not very favorably disposed to Akbar. Besides, we have official farmans, regional histories, Jain narratives, Vamshavalis, and inscriptions from all over India.
Not one, I repeat, not one of these sources tells us a single story of Akbar and Birbal.
Sure, there was a courtier of Akbar by the name of Birbal. But he was not known at all for his wit. The first occurrence of any hint of Birbal's 'wit' is from an 18th-century biographical narrative named Ma'athir Al Umara. It is separated by more than 200 years from Birbal and was written by a person of Decani origin who had access to the tales of Vijayanagara emperor Sri Krishnadevaraya and his witty minister Tenali Ramakrishna.
The stories of Krishnadevaya and Tenali Ramakrishna were famous throughout the Deccan and South. The Deccani Urdu poets had access to these tales. After Aurangzeb's invasion of Golkonda in the 17th century, there was a massive exodus of Urdu poets from Deccan to the North. They brought with them the stories of Sri Krishnadevaraya and Tenali Ramakrishna. These stories were recast in a Mughal frame. Akbar took the role of Krishna Devaraya. A brahman, Birbal was an ideal representative of Tenali Ramakrishna. And thus the stories of Akbar and Birbal became prevalent throughout the 19th century into the present day.
No wonder then, that many stories are direct borrowings. Like the story of the honey pit or the story of elephants.
For more information, please read Meenakshi Khurana's works on Medieval India