I agree with @Bill's comment - to me, Anglophones are normally native speakers only.
But for me, that's largely because I wouldn't normally refer to "an Anglophone" in the first place. I use the plural form specifically to include people from many different countries who speak English as their mother tongue. Anglophones are representatives of whole linguistic communities.
But by most dictionary definitions, many non-native speakers here on ELU qualify as Anglophones (they speak English), so I can't deny that technically speaking Reg Dwight, for example, is an Anglophone. But in most contexts where I want to convey that someone speaks/understands English [fluently], that's how I phrase it. I've no real use for a word meaning person who speaks English fluently - either by birth, or through extensive exposure/familiarisation later in life.
For most people (read, me), Anglophone either identifies those specific members of a multi-lingual population having English as their primary language, or it collectively identifies all native speakers. But you can't normally "learn" your way into either category; you have to be born into it.
Best Answer
A "native speaker of English" refers to someone who has learned and used English from early childhood. It does not necessarily mean that it is the speaker's only language, but it means it is and has been the primary means of concept formation and communication. It means having lived in a truly English-speaking culture during one's formative years, so that English has been absorbed effortlessly as by osmosis.
One can have been born and grown up in a country that lists English as one of its official languages and not be a "native" speaker. For example, Canadians from Quebec cannot automatically be considered native English speakers even though many speak English quite well; they were brought up speaking French as a first language and think in French (or Canardien, as I have heard unkind Parisians refer to it). But the rest of Canada does largely consist of native speakers of English.
Speaking "like a native" of any language means more than just knowing vocabulary and grammar. Many educated foreign speakers speak better formal English than, say, many Americans or British or Australians. But formal English is only one aspect of the language. Knowing instantly what slang means, what cultural references mean, how to reduce syntax to a bare minimum and still convey precise meaning — all these things, and more, are what constitute native speech.