I want to write my NPCs like they are in-game. Very 2-dimensional. Limited dialogue. 

That’s easy. 

But how do I do that and still have meaningful relationships?

How do I make these relationships between MC and basically simple-robots interesting?

Eventually the NPCs will breakthrough their limitations but in the mean time… they still need to be entertaining.

I’m currently thinking something between the people in the Equilibrium movie and like Ground Hog Day. 

How do you make 2d characters more interesting?

Make them 3d

You can make them live, but keep their personalities and attitudes like they are in a game. Keep their speech patterns as well.
"Welcome to Big Bob's, home of the Biggest Swords in all of Chaostown!"
Hi Bob. Why do you always say that?
"Why not? I certainly like saying it, and you can't ignore the fact that I have a Big Sword. Even my wife says that."
I need to sell some junk, how much for this stuff Bob?
"You came to the right place, my good friend. I can give you ..10 gold..for these rusty swords and dead animal skins...wait...you skinned a bunch of orcs?"
Orcs are hairy and I need the skinning experience.
"Say no more, say no more. Here at Big Bob's Big Swords, we buy it all. 8 gold."
You said 10 Bob.
"I also say I have a Big Sword. Neither were true, nor are these muskrat hides, but that's what I'll sell them as. But say, how about a deal?"
What's the deal, Bob?
"I'll take your fine rusty swords and muskrat hides and give you a ..BIG RUSTY SWORD!"
I'll take the 8 gold.
"Good dealing with you. I'll be here next time you have need of a Big Sword."
Bye Bob.

"The bandits are coming!"
"We must save my family!"
"Always remember to jump: Jump!"

In all seriousness though, just use dialogue from Oblivion or the Elder Scrolls and edit a bit.

Paradoxcloud Wrote: "The bandits are coming!"
"We must save my family!"
"Always remember to jump: Jump!"

In all seriousness though, just use dialogue from Oblivion or the Elder Scrolls and edit a bit.

I love you, you made my day lol.
I hate the fact that the SAO abridged version just kinda stopped.
So many good memories from that.

I think that  The Walrus King already touched and explained how to do it.
But it simply narrows down to make the characters, even if they are only 2d, interesting and enticing.

Let them have quests that are connected to them, and give them a backstory, that isn't fully explained to the fullest. 
Something like: 'I was once in the thieves guild until I was betrayed'.  Or 'My heritage goes way back to the first men of Valandor'. But make sure that they still stay in their own dimension.
And when they breakthrough their limitations the readers will be given more information about their backstory, which will turn them from the 2d to 3d.