According to Wiktionary, you can't use "a British" to refer to individual British people, though you can use it to refer to a race of people as a whole, but you can use "an Australian", and this matches what I already suspect.
I can't think of a good Google Ngram query to confirm this.
Is this the case? And if so, why is there a difference?
Those of us who live in these offshore islands often have some difficulty in describing ourselves. We can say that we’re British (although some residents of Northern Ireland may have a problem with that), but there’s no ready equivalent of the sentence She’s an Australian. She’s a Briton is just about possible, but sounds contrived. Britisher might also be found, particularly in films about WWII, but it’s not current.
The difficulty arises because British is an adjective, and only an adjective. Like certain other adjectives it can be used as the head of a noun phrase in some contexts. ‘The adjective-headed noun phrase usually refers to a group of people with the characteristic described by the adjective . . . The definite article is typically used with adjectives as noun phrase heads’ (‘Longman Student Grammar of Spoken and Written English’). This means that we can speak of the British, just as we can speak of the elderly and the poor, but we can no more speak of *a British than we can speak of *an elderly or *a poor.
You can't say a British: British is an adjectival, Briton is the demonym.
The demonym describes the person from a place. So:
The adjectival can be used as an adjectival noun, or an adjective:
Usually adjectivals and demonyms are the same, but there are exceptions.
-ish adjectivals have irregular demonyms, -ese adjectivals used as demonyms sound old-fashioned ("He is a Chinese"), and others are truly exceptional: French to French[wo]man, etc.
'Ancient Britons' is a recognised term in history and archaeology referring to the original inhabitants.
Brits is used but can often have negative associations, eg the Brits behaving badly abroad, getting into drunken fights with football supporters. I would always refer to someone as British and not a Brit or Briton.
Other countries have a similar adjectival word but for some reason adding 'man' or 'woman' doesn't sound as clumsy eg Frenchman. The individual countries of the UK will happily call their citizens an English/Irish/Scots/Welshman/woman but I pretty sure I've never seen Britishman/woman written all as one word.
Which still leaves and unsatisfactory gap if you wanted to say - "I shared a car with a German, an Australian and a ....?"
