Morphy defense in the Ruy Lopez

After 1.e4 e5 2.Nf3 Nc6 3.Bb5 a3, why is the recommended move by stockfish 4.Ba4, instead of 4.Bxc6 since the whole point of the Ruy Lopez is to attack white's only defender of e5? Isn't e5 a hanging pawn after black recaptures the bishop?

Well first of all, I think 4. Bxc6 is definitely a playable move. Many strong players have tried it. But they don't do it to win a pawn because it doesn't win a pawn.

So for instance after the line you gave with 4...dxc6 5. Nxe5 black can choose either Qd4 or Qg5 and in both cases black just wins the pawn back instantly by making a fork.

If 5. Nxe5 doesn't win a pawn, then it doesn't make sense for white to play it, because then the position just opens up and black's bishop pair is gonna become more of a factor. So usually the Bxc6 lines are a bit more positional than that. But anyway

4. Ba4 keeps the idea alive of *potentially* taking on c6/e5 later. Maybe it could be a thing someday, even if it isn't right now. Like when white castles, it could be a stronger idea because then if black goes after white's e-pawn in return white may have d4/Re1 and expose the open e-file towards black's king. Something like that. It's complicated but the point is 

a) Bxc6/Nxe5 right away does not win a pawn

b) strong players, such as Stockfish, sometimes prefer keeping the tension (and the bishop pair)

This of course doesn't mean that Bxc6 in itself isn't a very good move that any master could have as a part of their repertoire with success.

The e5 pawn is not hanging after 4. Bxc6 because of a winning tactic for black.

The exchange variation 4. Bxc6 is playable but it will not win a pawn. It is considered more passive and is basically totally equal. 4. Ba4 maintains the bishop pair and the advantage.

White's idea is only to take on c6 when the e4 pawn is defended so that Nxe5 does win a pawn. This is why whenever white defends the e4 pawn, black immediately plays b5. If we observe the main line, we can see this.

6. Re1 b5.

In the position after 5. 0-0, it appears that black can win a pawn with 5. ...Nxe4. However, white will win back the pawn with 6. d4. Black can't really play 6. ...exd4 because of 7. Re1 pinning the knight and black is in trouble. This is known as the Riga variation. It gets complicated but the final evaluation is much better for white.

5. ...Nxe4 is a playable and respectable variation known as the Open Ruy Lopez and black simply plays 6. ...b5 7. Bb3 d5 instead of 6. ...exd4.

Yeah I'm kinda a fan of 4. Ba4 Nf6 5. 0-0 Nxe4 for black, relatively. If I played 1...e5 these days I think the Open Spanish would be my choice. I've recommended it to students. But obv white doesn't lose a pawn. It's a good example of the open e-file thing I mentioned in my comment. White gets pawn back then game goes on

Yes, I like the Open Ruy Lopez. It's a very active defence for black.

I also like the Closed Zaitsev and the Arkhangelsk.

I play the Sicilian but I've considered developing an e5 repertoire as a secondary weapon. These are the 3 defences I'm thinking of against the Ruy.