In addition to the other answers, I'd like to add that the Schengen area, the EU, and the Eurozone are all technically separate, none is a subset of one of the others:
Ireland and Cyprus are in EU & Eurozone but not Schengen; Poland, Hungary (and more) are in EU & Schengen but not Eurozone; Switzerland is in Schengen but neither EU nor Eurozone; Montenegro and Kosovo are in the Eurozone but neither the EU nor Schengen.
In europe we have a kind of mini passport, called person id. Which only works in your own country and other shengen countries. It’s nearly the same cost as a passport (at least in my municipality)
You are required to have a passport (or id) with you (as in, that’s what the law says). Even in your own country. But in your own country a drivers license is usually also sufficient.
But in practice you will almost never be asked to show any of those. In your own country, nor abroad.
That depends on country, in Poland you don't need to carry any ID on you anymore (you're then required to remember PESEL number, and recite it to police if asked; 11 digits, six of those are birthday).
In practice you can get away with not having ID on your person in most countries as long as its reasonably close by. Technically you could get in trouble though so better carry one if you might provoke the police.
https://en.wikipedia.org/wiki/Template:Supranational_Europea...
https://en.wikipedia.org/wiki/Schengen_Area
https://en.wikipedia.org/wiki/European_Union
https://en.wikipedia.org/wiki/Eurozone
Ireland and Cyprus are in EU & Eurozone but not Schengen; Poland, Hungary (and more) are in EU & Schengen but not Eurozone; Switzerland is in Schengen but neither EU nor Eurozone; Montenegro and Kosovo are in the Eurozone but neither the EU nor Schengen.