Well for starters, Bxb6 is a bad move for white because the bishop on e3 is a key piece in white's position. Notice that all of white's pawns are fixed on light squares. If you give up the dark squared bishop so easily, then you'll get killed by a knight on c5, a bishop on f4, a queen on h4, etc. Moreover, your remaining bishop currently on f1 becomes a lot worse if you can no longer push the light squared pawns to dark squares. So Bxb6 is unthinkable unless you're winning something serious immediately (I wouldn't even do it if I was winning a pawn).
Finally, you say "white will create a doubled pawn weakness" with this move, but doubled pawns are only bad if they can become targets. Notice how the b file is closed, so no rook can ever attack those pawns. White just traded their dark squared bishop, so no way to attack the b pawn with a bishop either. Actually, with the semi-open c file, now it's black's turn to target the backwards c4 pawn. The doubled b pawns actually give you even more avenues for your pieces than if the pawn remained on c7!
Now are there any positives to Nb6 in of itself? Sort of. One problem in black's position is that there are too many pieces for too few squares. Both of your knights want to use c5 for instance. Your bishop and knight want to use d7. etc. So black is truly looking to get in some trades---any trades. Nb6 allows Bd7+Nc5+Nba4 finally trading the b6 knight for the c3 knight. If this can be achieved, then black finally has solved their lack of space issue. Is this slow? Incredibly slow. This is probably why the position is not so good for black overall. However, in closed/locked positions like this, you have more time for maneuvers like this, so the plan is possibly not too slow.
Natural moves from white will therefore be too slow in response to this plan. The real question is: can white do something scary on the kingside, while black uses a lot of tempi on the queenside to achieve a simple piece trade?
[FEN "r1bq1rk1/1pp2pbp/nn1p2p1/p2Pp3/2P1P1P1/P1N1BN1P/1P3P2/R2QKB1R w KQ - 1 11"]
1. Rg1 (1. Bxb6? cxb6 {Now black has complete control over the dark squares in the position. With Nc5 and/or Bh6 coming next, black is already probably for choice.}) Bd7 2. h4! {An important move - going for black's king immediately.} Nc5 3. a4! {Taking one time out to stop black's maneuver. White is for choice now that black's idea has been stopped, and the kingside attack rages on with g5+h5+etc} (2. Be2?! Nc5 3. Rb1? {Better than Rc1 which will run into an eventual fork, but simply too slow. Black now achieves everything they want for free.} Nba4! 4. Nxa4 Bxa4 5. b3 Bd7 {and black is for choice with an uncramped position.})
