I felt the authors skirted around a key question, namely what the nature of European/Western identity is. We might expect a book with such a title to address this. We have to make our own inferences. Towards the end of the book, they report a very interesting Latin inscription recently discovered in London, dated to "the late second century AD".
To the divine will of the emperors
This dedication was set up by a man carrying a good Roman name: Tibernius Celerianus, a native of Beauvais (ancient Caesaromagus Bellovacorum) in northern Gaul. Celerianus describes himself as a "seafarerer of the Londoners" (moritix Londiniensium), and we should probably understand him to be the agent of a shipping company which transported goods between London and northern Gaul.
It is very striking that Celerianus chose to define himself with the curious term "moritix". Moritix is not a Latin word at all, but an ancient Celtic term meaning "seafarer". There is, of course, a perfectly good Latin word meaning exactly the same thing (nauta). Why did Celerianus choose to use the old Celtic word? Was he trying, consciously or unconsciously, to emphasize his local Celtic identity?
The real cultural affiliations of a man like Celerianus are desperately difficult to recover: a native of northern Gaul, with a Roman name; worshiper of a superficially Romanized Celtic deity of his native region, but also of the reigning Roman emperors; capable of setting up a dedicatory inscription in impeccable Latin, but opting for a local Celtic term to describe his profession.
So let's say Celerianus retained a Celtic-Gallic ethnic identity. We might also go so far as to say that the authors discuss this inscription as much as they do because Celerianus is a quintessential "European". This implies that "European" unites two strands of identity and worldview, the ethnic (a set of specific local, related ethnic identities, generally harkening back to a "heroic barbarian past") within certain political and cultural superstructures and philosophical traditions. This Celerianus personifies it. Seems reasonable.