England first became a country, to all intents and purposes, under the rule of Alfred the Great. The uniting for formerly separate kingdoms against the Viking invasions, plus the rise to power of the single royal house in this period meant that this was the first time that English people began to define themselves as one people against a threat from overseas, where formerly they had defined themselves against the other peoples living on the same island as them.

Note that I say English. People living in moden Wales and Scotland had retained their Celtic makeup in this period, and would still not identify as Anglo saxon. That primarily applies to the people living in modern England, where the migration of germanic people shortly after the Romans left left a significant mark on the culture and genetic makeup. While it is difficult to estimate exactly how much these migrations affected the ethnic makeup of England, the cultural impact through the rise of germanic families to royal power made a big difference to identity. It was therefore these ethnic groups after which English people named themselves when they looked back to figure out where their nation came from.

I wrote my dissertation on this - I'm on mobile right now but I can post later and it should cover your question and provide references and further reading if you are interested.