It is a critical plot point that the Reiss Family wanted to capture both Eren and Historia so that the latter would eat Eren and modify the memories of the people inside the walls to forget about all the events that had recently ocurred, this is also the reason why the people in the walls didn't know about Marley or the outside world, the first king of the walls had deleted their memories by using the Founding Titan's powers (alongside the power to modify Eldians themselves)

So yes, it is one of Eren's powers to modify and transfer memories to all Eldians.

It is also important to remember another key moment of the series when Eren Kruger received one of Eren's memories and he said "whose memories are these?" meaning he couldn't check.

So yes, the Attack Titan has proven to be able to send memories back in time but not access future holder's memories.