Maz Kanata was born over a millennium before the conflict between the First Order and the Resistance. So she is very old and would have known about the goings on because she housed many smugglers throughout the years.
I believe she meant not the exact same physical eyes but she can tell what the person is thinking and feeling eg determined eyes. She is also force sensitive, so not only could she know the intentions/thoughts/feelings of people by looking in their eyes (“The Eyes are the window to your soul”) she can use the force to probe further.
The empire along with emperor Palpatine would have been common knowledge among anyone living under a planet of his rule or the surrounding systems. Darth Vader being his right hand and also the face of the imperial army would be just as know if not more. Vader was not afraid to show his skills both with the dark side and his lightsaber. Maz being as old as she was and having lived through so much would easily have know Vader was a Sith. She might not have known Palpatine was a Sith but it would not have been a wild guess on her part as he would complete the rule of two.
Kanata spent centuries traveling the galaxy, collecting antiques and trinkets which she kept inside the storehouses and vaults of her castle. One such item was the lightsaber crafted by Jedi Knight Anakin Skywalker and later wielded by Skywalker's son, Luke, who was instrumental in toppling the Galactic Empire.
-- Star Wars Wikia
There's no canon answer to the specifics on how she got it but lightsabers being so rare it's not a wild guess for her to believe it was Luke's. Her being force sensitive and the lightsaber obviously still connected to Luke, could have also helped her to verify it's previous owner.