But why? It all goes back to a rather strange use of 'bottle' to mean 'bravery' or 'nerve', which has been around for nearly a century now. So if someone has lost their bottle, they've lost their nerve, they're afraid. The verb 'to bottle' soon followed: you could 'bottle out of' something, or simply 'bottle it', if you didn't have the guts to do it. And so we got 'bottler'.
But the original question why remains. There's an old slang expression 'no bottle' meaning 'no good' which may have something to do with it, and it's often claimed that it's linked with Cockney rhyming slang 'bottle and glass'. That stands for 'arse', and various not entirely convincing attempts have been made to connect that with the idea of courage.
And a word of warning: in Australia, 'bottler' means 'someone or something excellent' (as in "That try he scored was a real bottler"). A slippery thing is slang.