Geography is fairly obvious. History certainly has an important role to play in modern Europe. Borders have changed, people have moved, monarchs, nobility and ordinary folks have intermarried, ideas have circulated, trends and artistic styles have been shared...
Religion seemed to have played a cohesive role from the Middle Ages onwards, but Christianity cannot really be seen as European per se. It is a Middle-Eastern religion which has spread on all 5 continents and now has much more followers in the Americas or Africa than in Europe. In fact Europe has been more divided by religion (Protestant vs Catholics vs Orthodox vs Jews, and now vs Muslims) than unified by it.
I think what makes Europe Europe is its historical and cultural heritage. North America, Australia and New Zealand clearly belong to the European civilisation, share the same roots. But it's not Europe because it lacks the historical buildings and historical intrigues in almost every single village. The lack the century old traditions, local dishes that go back to the depth of time, and the common memory of European history.