Green’s Dictionary of Slang

touch n.²

(Aus.) style, fashion, manner.

1915		C.J. Dennis Songs of a Sentimental Bloke gloss. 🌐 Touch – Manner; mode; fashion.
1918		Aussie (France) 4 Apr. 9/2: So when another draft blows in from Blighty / And reinforcements lob into the line, / Don’t make the Anzac touch too high and mighty; / Don’t think of them as frozen-footed swine.

In phrases

come the touch on (v.)

(Aus.) to adopt a manner or attitude.

1984

Partridge DSUE (8th edn).