What is the difference between ‘Is it free’ and ‘Is it on the house?’

"Free" and "on the house" both mean that you don't have to pay, but the inferred meaning is slightly different.

If something is "free" it is without charge. For example, you might receive a voucher through the mail that says you are entitled to a free drink if you hand the voucher in at a bar. That drink was never going to have a charge attached to it, because it was offered by the bar as a promotion.

When a drink is offered "on the house", the inference is that the "house" or establishment is buying the drink for you. Similarly, if you were drinking with a friend and he/she said "these are on me" it would mean that they are picking up the bill.

True, the outcome of both is essentially the same - a drink that you don't have to pay for. It is also true that some establishments may use the two expressions interchangeably without any attention to the root meanings.

As to whether or not it is a known idiom in Australian English I would have to say yes on the basis that it appears in the script for Australian movie Crocodile Dundee II:

-G'day, mate! Bet you've got some stories to tell, eh?
-Have I ever, son. Have I ever. But not on a dry throat.
-Ida?
-The drinks are on the house.
-That's a first!

It is an AmE expression, probably not common in Australia;

From the Oxford English Dictionary: "on the house: at the expense of the tavern, saloon, etc.; also transf. and fig. (orig. U.S.)

1889 Kansas City (Missouri) Times & Star 30 Nov., The first drink Thursday was 'on the house' in the leading saloons.

(The Phrase Finder)

Free drinks (on the house) and (free drinks are) on the house in the context of a bar, pub, restaurant, business etc. mean the same thing

At the expense of the establishment, as in

• This hotel serves an afternoon tea that's on the house.
  This idiom uses house in the sense of "an inn, tavern, or other building serving the public." [Late 1800s]

The expression is very common in the UK, so I would imagine that it's also familiar to Australians; however, in New Zealand and in Australia a term more commonly used to mean “at one's own expense” is shout

2. NZ Australian informal with two objects
Treat (someone) to (something, especially a drink)

‘I'll shout you a beer’
‘He'll happily let you shout him a drink and not return the favour, for example.’

2.1 no object Buy a round of drinks.
‘anyone shooting a hole in one must shout for all players present on the course’

(Oxford Dictionaries)

The awkward atmosphere among the group of friends could have been provoked by the speaker's assumption that he or she didn't have to pay, either the establishment was offering its clients something free of charge or someone in the group had offered to pay/shout for a round (of drinks).

References on EL&U
What terms describe who pays for a meal?
Can I “shout someone out”, or only “give a shout out to” someone (to recognize them)?