No, the distinction is about a person's feeling of national identity. Technically anyone from Great Britain (England, Scotland and Wales) is British, but many people feel a closer connection to the particular nation they're from. Your fiancé apparently feels more of an English identity. My fiancé is Scottish, and especially since he is a nationalist who wants Scottish independence, he does not refer to himself as British and doesn't like other people referring to him that way. He realises that he is a citizen of the UK, but when asked his nationality he will always put Scottish.