I would say Black will be trying to play a Queens Gambit Declined structure, but White is trying to play some form of KIA/Reti so the position is not definitively classified for Black other than D02: Queen's Pawn Game
Which opening is this for black?

I would say Black will be trying to play a Queens Gambit Declined structure, but White is trying to play some form of KIA/Reti so the position is not definitively classified for Black other than D02: Queen's Pawn Game
It is not a QGD (no c4) nor a Reti.
If you look at any d4 sidelines book or anti-king's Indian book, you will see that lines where White plays d4 with no c4 and develops the Bishop to g2, they simply call this the Kingside Fianchetto, which is basically any position where White plays d4, g3, and Bg2 with specifically no c4. If c4 is played, you wind up, depending on what Black plays, in a Fianchetto King's Indian, Fianchetto Benoni, Fianchetto Grunfeld, Catalan, or Dutch.
it's a common formation used by queen's gambit declined players, except i don't see the need for h6. b6 preparing bb7 seems more harmonious. white is likely angling for e4, or some kind of pillsbury formation with ne5 and f4 and against that you would prefer the pawn to be on h7 as f6 is easier to play later.
#4
It is black to move. So 6...h6 was probably meant to prevent 7 Bg5 to trade away his bad bishop and prepare e4, so white played 7 Nbd2 instead.
#6
In this variation white can play 8 e4 dxe4 9 Nfd2
6...c5 is better than 6...b6: attacking the center. A bishop on b7 has no bright prospect staring at its own pawn d5.
6 c3 by white is passive: white should play 6 c4 and then it is a Catalan.
8 e4 de 9 nfd2 is creative but doesn't give white anything if black unpins with qe8
better in what regard? b6 does not preclude playing c5 later.
a bishop on b7 does not look impressive, but the point is to control the e4 square in order to try to place a knight there. it's not going to stay there forever, nor is white going to be able to exchange everything off leaving any good piece versus that bishop. besides white has to refrain from playing c4 or e4 in order to keep that bishop staring at d5, and if so then what exactly does white play for as a pawn break?
i think the comment about the bishop on b7 is rather dogmatic
From black's perspective, what opening is this? It a Queen's pawn game, maybe some kind of Indian, but beyond that I don't know.