Based on The Inheritance of Rome I'd suspect that most people would simply understand them an Roman buildings. The author (Whicham?) Makes the case that most of the areas previously under Roman rule would have continued considering themselves Roman for quite some time even as the scope of the empire's power shrank.

In large part he paints the transition and fall of the empire into essentially the rise of localized power holders taking over the taxation and bureaucracy in their regions and slowly ceasing ties to Rome as Rome's ability to muster men or resources to influence the provinces fell off.

So in this regard I don't suspect there was some massive loss of consciousness of the Roman Empire. In fact Rome existed continuously as a city and retained prominence as the base of a new sprawling bureaucracy - the Catholic Church. In fact the church was one way in which normalcy transitioned from pre to post Roman Europe as these networks and hierarchies were still in place.

If you are interested in this period I highly recommend the book. It does a wonderful job starting just as the Roman Empire is really falling and traces the next 5-600 years in the west and east to show how both regions slowly evolved into the middle ages.