My comments:

House here is a colloquialism for the café, bar, pub or restaurant or whatever such building (house) you’re in – where you wine and dine.

When the drink is “on the house”, it means the owner of the place pays for it out of his own pocket, by way of showing extra hospitality.

That is to say, the cost of the drink is put “on the house”, not on you.

To wit, you have it for free.

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