There are a lot of reasons why he does it.
Barry hugs Oliver because he knows Oliver isn't that kind of guy, but he is, and sometimes he just wants to be a little selfish. Sometimes he wants to take what he can get because he doesn't know when he'll be given another opportunity. He's learned that it's easy to believe there will always be a tomorrow when one might not exist; luckily, he's also learned that, in the times he needs it most, he's given an opportunity to make things right.
He also hugs him because the Arrow is the only superhero he's met who looks at the Flash with admiration and affection instead of contempt or confusion. He treats the Flash like an equal, and coming from a guy like the Arrow? That says a lot. Especially when Barry knows exactly how precious Oliver's affection is, how exclusively he gives it out. To be in that circle of trust means the world to him.
But mostly he hugs Oliver because he is there, and Barry wants to, needs to, and no amount of eye-rolling is going to stop him (he'd use the super speed if he had to because even Oliver couldn't stop him if he did, but it wouldn't be the same, hugging a stiff body, hugging someone who didn't want to be hugged). He needs to hold someone. He needs to feel like there's a real person under the Arrow's mask who can help him stop cataclysms from destroying the world.
He needs it to be real, to cement the fact that they did change the future, together, and maybe he shouldn't have told Oliver, maybe it was foolish and selfish and is going to have catastrophic consequences – but Oliver is right. What could possibly be worse than everyone he knows and loves being incinerated, the city he loves burned, the world turned utterly upside down where his only prayer of staying alive is to run, forever, until there's nowhere left to go?
Standing in the center of city with the world's coolest vigilante, Barry lets out a breath as he hugs Oliver as tightly as he can, willing it to be impressed on his skin for those nights when he'll need it, when he'll need someone's support, someone who's been at this a lot longer than he has and can keep him alive when everything else is crashing down. He hopes that the way Oliver hugs him back tightly squashes the tiny, residual portion of his mind that says, You'll never be good enough to stay on his level.
He doesn't need to be as good as the Arrow – just good enough.
When he thinks of the catastrophe they've avoided, and the catastrophe he avoided on his own so long ago, he feels the strength slip out of his knees, a panic beyond words sweeping over him that a future so dark could actually happen.
So the hug is – a way of reminding himself that the tangible, the here and now, is safe, okay, and there's no tsunami, there's no heat wave, and everyone is alive and going to stay that way.
He lets Oliver go after six thousand milliseconds, savoring each one until at last Oliver's shoulders and arms relax and they step back from each other.
And he knows, in some grateful, relieved, childlike corner of his mind, that as long as Oliver is alive, everything is going to be okay. Because Oliver always finds a way – he's smart, and resourceful, and he does it without powers, going up against the incredible and emerging unscathed – and Barry can't describe how much it means to have a guy like Oliver on your side.
It's like having a big brother. He kind of likes it.
A grumpy, could-kill-you-with-one-look, forgets-to-restock-the-beers kind of brother, but Barry's faced down worse and alcohol doesn't do it for him anyway.
As they go their separate ways at last, Barry vows to uphold his end of the unspoken promise.
If you ever need me, I'll be there.
As fast as I can.
And as he watches Oliver walk away he can't help but grin, because God, he is so, so grateful this absurd, wonderful human being exists.
Luckily for him, he has two very unreserved friends to hug when he gets home. Even if he misses Oliver's strength and tangibility already, he consoles himself with the thought all the way back.
(Cisco really is a great sport about it.
"Are we cuddling?" he asks, because Barry's just sort of sprawled on the couch next to him, and he's not hugging him, exactly, but he is all over him, his side pressed against him, eyelids drooping as he watches Cisco play QuizUp on his phone, cheek tucked on his chest.
"Mm, nope. We're sitting. On a couch. Very close to each other."
It's total bullshit and Cisco sees right through it, but he just sighs dramatically and says, "Uh huh."
Then Caitlin's home and she just sighs before sliding into the free space beside Barry and wrapping her arms around his waist to cuddle into him, and the shift in events is so startling Cisco asks, "Are you okay?"
"Yes," she replies, and Barry can feel it against his shoulder, and the way Cisco's relaxed and Caitlin's relaxed too make him calmer, like everything really is going to be okay.
He lets it go on for as long as they let him, doesn't release Cisco or push Caitlin off, and when his eyelids droop shut he lets that happen, too, savoring the fact that he has friends, and they're alive, and everything's going to be okay.)