Board Thread:Series general discussion/@comment-5088097-20170624223327/@comment-2112031-20170625121936

Here's what we're told in Assassin's Creed during the tutorial:

"Now that you are fully synched, a new ability is available to you. Referred to as Eagle Vision, this sixth sense helped your ancestor understand the intentions of people around him."

It's pretty clearly stated to be an ability that Altaïr actually possessed, rather than being an interpretation of the Animus.

As for Bayek, I suspect that the explanation we're gonna get is that he has far more Isu DNA in him than later generations of Assassins, thus giving him access to a far more powerful (and literal) Eagle Vision.