Do they check passports between countries in Europe?
Although in most European countries there are no passport border checks, companies can refuse to board you if you do not have a travelling passport (or some other ID proving that you are in Europe legally).
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There is one single EU external border. This means, for instance, that there are no longer any border checks at the Dutch borders with Belgium and Germany.
What appears on their screen is basically all the information they have about you. All your travel history to and from that country. The consular officer's note when you were interviewed. Everything written in your passport, including identifying details and any travel history it contains.