Proper adjective, or noun with Indefinite or zero article, when defining national identity?

Actually a US citizen can say "I'm an American". In fact there is a George Gershwin composition, a song and a 1951 film all called "An American in Paris" but 'American' used as a noun in this way is actually a shortened form of "American person" or "American citizen" (and 'American citizen' is, strictly, a synonym for "citizen of the United States of America" since Canadians, Mexicans, Brazilians, Agentinians and inhabitants of all the other countries in both American continents are Americans in one sense).

If we say "I'm an X" referring to our nationality then X has either to be a noun or an adjective accepted as a noun (as American is). Only certain adjectives have acquired this status and 'British', 'Polish' and 'French' are not among them.

It is actually arbitrary whether an adjective has this status, most of the ones that end in 'an' (Jamaican, German, Russian, Australian for example) and 'i' (Israeli, Saudi, Pakistani and so on) are acceptable but those ending in 'ish' are not. I have no explanation for this, it's just the way the language has developed.

You will find, though, that nationalities whose adjectives are not acceptable as nouns tend to have specific nouns for their citizens, even if they are informal. Think of "I'm a Frenchman", "He's a Brit", "She's a New Zealander (or Kiwi)".

Edit After thinking about this a bit more I've realised that the convention ruling when national and ethnic adjectives are acceptable when acting as singlular nouns is probably related to whether such nouns are used as plurals in other cases. For instance we say "The British have bought a lot of properties in Spain" which means that 'British' as a noun is considered to be plural (or perhaps a mass noun) so "a British" is unacceptable. However we say "The Americans landed on the moon" using the plural form of the noun so, when we are talking about one person, we can say "The man in the Bermuda shorts is an American" because there is a singular form of the noun available.

Things are complicated for the UK (warning - do not read if you value your sanity!)

The UK consists of several countries. These are England, Scotland, Wales and Northern Ireland. However Britain includes only three of those countries, England, Scotland and Wales

Unfortunately the rules aren't consistent. One can say, "I'm a Scot", but this does not work for the other countries. We do not say, for example, "I'm an English".

The safest way out of this is to use

She's English

He's Welsh

She's Scottish

He's Northern Irish

This becomes even more complicated when we talk about Britain. Britain consists of England, Scotland and Wales but not Northern Ireland! So people from the mainland can say

She is a Brit
This is informal but perfectly acceptable in almost all contexts except official forms and academic writing.

He is British
This can be used formally or informally

So what about the UK?

No-one says UKish!

The only correct forms are

He is from the UK.

or, formally

She is a UK citizen.

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Conclusion

If in doubt, say

She is English, Scottish, Welsh, Northern Irish, British.

or say

S/he is from [name of country], e.g.

He is from England/Scotland/Wales/Northern Ireland/Britain

but

She is from the UK.

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Good luck but don't worry if you get it wrong. If you are not a native speaker, no-one will mind.