Since a game with all 14 pieces (as kings our excluded) hanging has probably not happened, and will not for a long time, I will answer as best as I can.
If a game with all 14 pieces haning is not out there, I think that the best we can do is to find a postion with the most haning pieces as possible. I do have an answer, and it may not be the best, but you are welcome to find improvments if you can.
I found a game that has, on the board on move 23 there are 6 hanging pieces on the board by your defintion. Black can capture 3 White pieces, and if the turn switched, White could also capture 3 Black pieces.
[Title "Loal W Davis-Magesh Chandran Panchanathan, HB Global Chess Challenge, Minneapolis, MN USA, 5/18/2019"]
1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5 Na5 6. Bb5+ c6 7. dxc6 bxc6 8. Be2 h6 9. Nf3 e4 10. Ne5 Bd6 11. f4 exf3 12. Nxf3 O-O 13. d4 Rb8 14. a3 Re8 15. O-O c5 16. Nc3 cxd4 17. Qxd4 Nc6 18. Qh4 Bf5 19. Bd3 Bc5+ 20. Kh1 Bxd3 21. cxd3 Qxd3 22. Bxh6 Rxb2 23. Na4 Rc2 24. Nxc5 Rxc5 25. Bg5 Ne4 26. Bf4 Rc2 27. Ng5 Nxg5 28. Qxg5 Nd4 29. Rfe1 Rce2 30. Rxe2 Qxe2 31. h3 Nc2 32. Rc1 Qe4 33. Bg3 Ne3 34. Bf2 f6 35. Qxe3 Qxe3 36. Bxe3 Rxe3 37. Ra1 a5 38. Ra2 a4 39. Kg1 Kf7 40. Kf2 Rc3 41. Rb2 Rxa3 42. Rb7+ Kg6 43. Ra7 Ra1 44. Kg3 a3 45. Kf2 Kh6 46. Ra5 g6 47. Kg3 Ra2 48. Kf3 f5 49. Ra7 Ra1 50. Kf2 f4 51. Ra8 a2 52. Ra5 f3 53. Kg3 Kg7 54. Ra6 Kf7 55. Kf2 Ke7 56. Kg3 Kd7 57. Kf2 Kc7 58. Ra3 Kb6 59. Ra8 Kc5 60. Rc8+ Kb6
However, if you want a positon that is “pure,” where all pieces are hanging, I found this beautiful game where all 5 pieces on the board are hanging at the end of the game.
[Title "Veselin Topalov-Alexander Morozevich, NAO Masters, Cannes FRA, 2/22/2019"]
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O b5 6. Bb3 Bc5 7. a4 Rb8 8. c3 d6 9. d4 Bb6 10. axb5 axb5 11. Na3 O-O 12. Nxb5 Bg4 13. Bc2 h6 14. dxe5 Nxe5 15. Nbd4 d5 16. exd5 Qxd5 17. h3 Bh5 18. g4 Nfxg4 19. Nxe5 Nxe5 20. Qxh5 Bxd4 21. Qf5 Nf3+ 22. Kh1 Qxf5 23. Bxf5 Bf6 24. Ra7 c5 25. Be4 Ne5 26. Rd1 Nc4 27. Bb7 Bg5 28. Bxg5 hxg5 29. b3 Ne5 30. Rd5 Rfe8 31. Rxc5 Re7 32. Rb5 Nd3 33. Be4
Feel free to hit me up with any improvements on these numbers.
On a semi-related note, the record for the longest time a piece was “hanging,” is the below game in which the Black knight on g4 could have been captured by the White h-pawn for 27 consecutive moves. Tim Krabbe has it in his chess records of course.
[Title "Anthony Miles-Ljubomir Ljubojevic, Interpolis 10th, Tilburg NED, 1986"]
1.Nf3 c5 2.c4 g6 3.e4 Bg7 4.d4 Nc6 5.dxc5 Qa5+ 6.Nfd2 Qxc5 7.Be2 d6 8.O-O Nd4 9.Bd3 Nf6 10.Nb3 Nxb3 11.axb3 Ng4 12.h3 h5 13.Qd2 Qe5 14.f4 Qc5+ 15.Kh1 Bd7 16.Nc3 e6 17.Qe2 Bd4 18.Bd2 Qb6 19.Nd1 a6 20.Ba5 Qa7 21.Bb4 Qb6 22.Ba5 Qc6 23.Qe1 e5 24.Bb4 Qb6 25.Ba5 Qa7 26.f5 gxf5 27.exf5 Bc6 28.Be4 Kd7 29.f6 Rag8 30.Bxc6+ bxc6 31.Qe4 Qb7 32.Bc3 Ba7 33.b4 Kc8 34.c5 dxc5 35.Qf5+ Kb8 36.Bxe5+ Ka8 37.bxc5 Bxc5 38.Bg3 Ba7 39.Qd3 Ne3 40.Rf3 Rd8 41.Bd6 Nxd1 42.Qxd1 Rhg8 43.b4 Rg5 44.Rd3 c5 45.Bg3 Rgd5 46.Rxd5 Qxd5 47.Qa4 cxb4 48.Qxa6 Qb7 49.Qf1 Rd7 50.Bf2 Qc6 51.Bg1 Rb7 52.Qf2 Qd7 53.Qf3 Kb8 54.Qf4+ Ka8 55.Qe4 Kb8 56.Bxa7+ Rxa7 57.Qxb4+ Ka8 58.Qf8+ Kb7 59.Rxa7+ Kxa7 60.Qe7