Is this same-color bishop endgame actually won for black?
Black has faster king, but he has permanent weakness in view of the
His bishop is bad and kingside pawns are on the same color he is. Although kingside pawn formation can be changed to suit the bishop, he will still be unable to attack the opponent because White can also change his kingside pawn structure to favor his bishop.
Furthermore, White king is fast enough to cut off opposing king with joined cooperation of his bishop and pawns.
Since Black king is faster, Black will be able to exchange weak
a4 pawn if this one becomes a problem, thus the position will result in a draw.
[fen "8/3b1pkp/6p1/8/p7/7P/1P2BPPK/8 b - - 0 1"]
1...Kf6 ( 1...g5! 2.Kg1 Kf6 3.Bc4! Be6 4.Bb5! Bb3 5.Kf1 Ke5 6.Ke2 Kd4 7.Kd2 Bd5 8.g3 Bb3 9.h4 f6! 10.Bd3! h6 11.f4! gxf4 12.gxf4 Bd5!? 13.Bc2 Bb3 14.Bd3= ) 2.Bc4! Be6 3.Bb5! Bb3 4.Kg1 Ke5 5.Kf1 Kd4 ( 5...g5!? 6.Ke1 Bd5 7.g3 Bb3 8.Kd2 Ke4 9.Bc6+ ( 9.Bd3+ Kf3 10.Ke1 h6 ( 10...h5 11.Be2+ Kg2 12.Bxh5 Kxh3 ) 11.Be2+ Kg2 12.Bf1+ Kh2 13.h4 gxh4 14.gxh4 f6! 15.Kd2! Kg1 16.Ke1 Kh2= ) Kd4 10.h4= ) 6.Ke1 Kc5 7.Be8! Bd5 8.g3 Kb4 9.h4! Kb3 10.Kd2 f6 ( 10...Kxb2 11.Bxa4 ) 11.Kc1 Bf3 12.Bf7+ Kb4 13.Kd2 Bg4 14.f4 Bf3 15.Bg8 h6 16.Bf7 Be4 17.Kc1 Bd3 18. Kd2 Bf5 19.g4 Bxg4 20.Bxg6 Be6 21.Bd3 Bf7 22.Kc2 Bc4 23.Be4 Bb3+ 24.Kd2 Bg8 25.Bd3 Bd5 26.Kc2 Bb3+ 27.Kd2 Be6 28.Kc2 Bf7=
No matter what Black tries to complicate the game, thus creating winning chances, we see that White can simply decline the offer and keep things simple which results in a draw.
Hopefully this answers your question. If you have further questions leave a comment.