I know my husband had a job once doing machine maintenance in a food processing factory and he had the option of shaving his facial hair or wearing a beard net (which went over his ears and covered his goatee), he chose the net but eventually shaved off his goatee as the net was annoying and got in his way.
I'd imagine that working somewhere like Aldi would mean that your DP wouldn't be able to wear a net as he would be in the public eye most of the time, so yes Aldi would be within their right to ask him to shave his facial hair off.
Not sure about the interview in Sydney though.