Was it normal for nobles to walk around dressed in their House’s colors and sigils all the time in real life the way it is in ASoIaF?

The Westerosi take it a bit far – unless they were in armor, most nobles wouldn’t wear surcoats with their coat of arms on it – but usually nobles would wear something more discrete, like a broach or cloak-pin or hatpin, and their servants would wear their livery.