Why was the time after prayer specified in verse 106 of Surat Al-Ma'idah for bearing witness, and is it permissible to bear witness at any other time? And why did the books of Tafsir (exegesis) state that the prayer referred to is the Asr prayer, and what is their evidence for that?

The Asr prayer is the middle prayer and the greatest of prayers. Its time is when the angels gather. The definite article "al" in "al-Asr" refers to a known entity. It is the prayer after which oaths were commonly taken by Arabs, either out of reverence for its time, or its proximity to sunset, or due to Allah's wrath upon those who swore a false oath during it. For this reason, some hadiths specifically warn against false oaths taken after it.