Unless you're in transit between countries, it doesn't affect you as much as you might think. I lived in Japan for a year and pretty much only used my passport when I entered and left the country. Same goes with my sisters who moved overseas. They take out their US passport about once every 4 years, when they come back to the USA to visit.

In Russia, it's actually super important since foreigners need to register at the local police station with their passport in any town/city where they state more than a few days. Not registering within the allotted time period, based on dates of transportation tickets like a train ticket or flight stub, makes you the target of police extortion.

Russia is definitely one of those "Ihre papiere bitte" countries.