What is the reason for specifying Hajj with the Day of Arafah, as stated in the Prophet's (PBUH) saying, "Hajj is Arafah"? And was the place named thus because Adam and Eve recognized each other there?

"Hajj is Arafah" because standing at Arafah is the most significant pillar of Hajj. Whoever misses standing there has missed Hajj, unlike the other pillars which can be compensated for. The reason for dedicating this day to standing (at Arafah) is among the matters that are hidden from the Muslim, and one must submit to it. It is appropriate to refrain from asking about the wisdom behind legislating such matters, as no benefit is hoped from doing so.