I think he was sharing the same kind of experiences as her. That conversation in the future where he tells her he was there to meet her, that knowing smile the whole time, and he realized he had to give her the information to tell him in the past.

No, not really. See, as a species, we have a universal experience, but we're divided by thoughts, ideas and culture. The interesting thing about it is that, it just takes one thing for people to find find common ground. Getting people to change their minds on anything is incredibly difficult. Once you find that primer, if you will, you have to power to bridge beings. Not change their minds so that they're connected to the motherboard, but to make them understand and see things differently. It's kind of like that scene in Batman v Superman where they're fighting, and Batman stops once he realizes that Superman is just like him, a man with a mother and a life.
"I wish I wasn't afraid all the time, but I am."
-V for Vendetta

Those words were his wife's last ones. I assume no one else could know them.this is why he listened.
Just forget you ever saw it. It's better that way.

well, he clearly says to her that she had accomplished something which his superiors could not, changing his mind.

I think he probably was already in the process of changing his mind because he was also having conversations with the leaders of the other affected nations.

I was under the impression that the other 11 were at a level of similar communication with their Abbott and Costello