Follow on question: how good/reliable/accurate were wanted ads and descriptions? How often were the wrong people brought in from far away only to be released when recognized as not the person being sought?
Common identification was rare until after the conclusion of the First World War in 1918 simply because the need for such mass verification was virtually non-existent. The drive for an easily acquirable form of identification began in the early part of the war, when national security became of paramount importance in both combatant and neutral states as the Allies and Central Powers grappled for control of Europe. In the period following the First World War, passports became an increasingly pressing issue as soldiers demobilized and were, in some cases, sent home from Europe to home destinations that could be half way across the world. Passports for personal international travel were finalized at the Paris Conference of 1920, but the standards required of every passport holder -- such as a passport picture and identifying marks -- rankled typically upper class travelers through the 1930s.
Earlier passports issued to diplomatic staff were usually only presaged by a common preamble by the issuing government. This led to the universal adoption of a request for protection that can still be seen today in many passports.
In the United Kingdom:
Her Britannic Majesty's Principal Secretary of State for Foreign and Commonwealth Affairs Requests and requires in the Name of Her Majesty all those whom it may concern to allow the bearer to pass freely without let or hindrance, and to afford the bearer such assistance and protection as may be necessary.
In the United States:
The Secretary of State of the United States of America hereby requests all whom it may concern to permit the citizen/national of the United States named herein to pass without delay or hindrance and in case of need to give all lawful aid and protection.
In the French Republic:
Nous, Ministre des Affaires étrangères requérons les officiers civils et militaires de la République Française et prions les Autorités des pays amis et alliés de laisser passer librement le titulaire du présent passeport et de lui donner aide et protection. ("We, The Minster of Foreign Affairs, require local civil and military officials and the authorities of our friends and allies to allow free movement of the holder of this passport and to give aid and protection [when needed].")
Domestic identification on a large scale took the form of internal passports, which were particularly famous in pre-revolutionary Russia in order to travel from place to place. In America, rather darkly, internal passports were exclusively the domain of slaves, who required them to travel even the shortest of distances. Domestic identification was therefore something to be avoided as a mark of crime or servitude; Victor Hugo cast Jean Valjean as the holder of an infamous yellow passport that reflected his status as a parolee.
While many nations had some sort of domestic passport to travel from town to town, in essence, a certificate of good behavior, there was no modern equivalent to a drivers license or identification card as we know it today. Your average American or Briton could therefore travel relatively incognito from, say, Ohio or Lancashire to Wyoming or London completely unimpeded save for regional markers such as accents, mannerisms or shibboleths that would mark them as an outsider.