When he saw the guy approaching the prison he didn't know yet he was Eli's nephew But by the time he had met him he had managed to find out After all, they are rich in records and they know where to begin their investigation: Prisioners in that jail, their relatives, their social network and these people's vehicles and well, yes, someone else said, and this could have been a factor, his car might have had data that could have made it all much faster.

I believe Coulson covered that with, "You worked hard to provide for your sister and her kids."

He knew the percentage of people that went to college from the guy's highschool, I think he might have found out about his two nephews.
"We're a team, Garrus. There's no Shepard without Vakarian."

Do we really need to be explained everything that doesn't happen onscreen? It's not at all implausible that Coulson found out the info with the super advanced computers he has access to. It's kind of like saying hey, where did Daisy get that leather jacket she wears now?
Yeah well, you know thats just like, your opinion man. - The Dude

Agreed.

Think so too. They also had a list of all people wotking at the reactor. Eli was on that list. The Ghostrider also knew about the reactor. A liitle investigation after his capture is needed from here on.

When he met the uncle the first time in prison, he didn't really know the connection, or it didn't matter.
After they captured Robbie, I'm sure he ran his prints, figured out his connection, and brought it up.

He knew for two reasons.

1. He's a SHIELD Agent (And Legit too!)
2. He's Coulson.
   I mean sure that doesn't explain it entirely, but that's pretty much all that is needed to explain it, because fiction.
   ILLYA KURYAKIN: Hey. Is My Name Illya?
   NAPOLEON SOLO: Who CARES?!

Driver's license, car registration, finger prints, his cell phone, etc