You are right that most of the time we refer to people from the UK as 'British'.
However, in fact, the correct term for a person from the UK is 'a UK citizen'. The UK is a political rather than a cultural entity. The UK is composed of Great Britain and Northern Ireland, so, strictly speaking, the term 'British' excludes citizens of Northern Ireland. People from England, Scotland and Wales are British, but people from Northern Ireland are not. All are UK citizens.
And to answer your second question - no, you cannot say 'He is a Chinese' or 'He is a British'. These are adjectives. They are not nouns.
You can say:
'He is British' or 'He is a British man/student/tourist'.
In general, the only nationalities where the adjective and noun form are the same are those ending in 'n' or 'i'. For example, you can say 'He's a Korean' or 'He's a Pakistani'.