Can anyone please explain why this position isn't a draw? I was watching this game on lichess.
Both of the players have the same resource i.e., the king and only a bishop. The bishops on the board are of the opposite color. But the game was continued and one of the players flagged on time. 

I ran a full computer analysis and the engine has shown the position as a draw. But, since the computer allowed the player to continue the game with that resource, my question is whether it is possible to win a game with "ONLY" a bishop, provided your opponent only has the same? If yes, then what should be the checkmate position? and will the result be different if your opponent has the same color bishop?

[FEN "8/1k1b4/8/2B5/8/K7/8/8 w - - 0 1"]
According to the FIDE Laws of Chess:


  5.2.2 The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position’.


The game position you show is not a draw because there are series of legal moves ending in checkmate for either side.

[FEN "8/1k1b4/8/2B5/8/K7/8/8 w - - 0 1"]

1. Kb3 Be6 2. Kc3 Kc7 3. Kd3 Kd7 4. Ke3 Ke7 5. Kf4 Kf7 6. Kg5 Kg8 7. Kh6 Kh8 8. Bd6 Bg8 9. Be5


Of course black's moves were stupid. That doesn't matter. They were all legal moves and so were all white's moves. A similar set of moves could be constructed to allow black to checkmate white. The sensible thing to do is to agree a draw in this position, particularly if there is an increment. 

If there is no increment and the player with more time wants to be spiteful then they can play on and try and flag the other player. Of course they shouldn't do this if they are playing Gata Kamsky GM FFL :-)

With same colored bishops this is not possible. The opposite colored bishop is required to block one of the king's escape squares.