I don't think it is so much of a surprise. The Roman empire enforced a lot of cultural norms and spread a common religion. The religion, in particular, was responsible for a largely common education and that still exists today! It is therefore no surprise that the member states that share that common education have a lot of political interaction. However, often that political interaction has been warlike and negative and it's not clear that one can peaceably enforce positive interaction between states.
The best thing that the EU could do to harmonise relations, in my opinion, would be to encourage an EU-wide educational system and make that education open to other, non-european states too. Take the harmonising parts of that ancient common religion and try to avoid the the mystical and manipulative hierarchies which caused a lot of bloodshed and which will, in my opinion, eventually bring it down.
