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The idea of using photos as a means of identification is almost as old a photography itself and seems to date back to 1876 in the USA.
Large expositions in London, Paris and elsewhere had all experienced problems with properly identifying employees, exhibitors, press and officials who made recurrent trips to the exhibition site....The solution at Philadelphia was what was called "the photographic ticket."
Source: Roger Hall, Gordon Dodds, Stanley Triggs, 'The World of William Notman: The Nineteenth Century Through a Master Lens' (1993)
The above source then quotes the Centennial's official report's description of this early photo ID:
The photographic ticket...in the form of a book-cover and engraved by a bank-note company.... On the front the number of the ticket, the name of the holder and his or her relation to the exhibition fully set forth....on the right hand side was set apart a space, of well-defined size, to be occupied by a photograph of the holder...
There's no mention of a signature here (nor is one evident in the picture of one such ticket shown in the Notman book), but this innovation seems to have started a trend, perhaps helped by the international make-up of the attendees (from Tunisia, Algeria, France, Germany, Japan, Turkey, Spain, Austria, China and other unspecified countries).
In the late 19th and early 20th centuries, cards with photos and sometimes signature of the bearer were used at exhibitions (see example here - Switzerland) and for press cards (see example here - France). However, these were not labelled 'Identity Cards' and seem to have been for use only at one particular event.
The earliest Identity Card with photo and signature of the bearer I've found is this 1911 example issued by the Austrian State Railway:
"Austrian identity card, Identity card, front and back". Source: Luminous-Lint
Photo cards with the bearers signature seem to have become increasingly common during World War I in France, Canada, the US and other countries for military personnel, though they did not necessarily use the words 'identity card' or 'carte d'identite' (see example here).
Identity cards were also issued to American Expeditionary Forces from at least 1919, of which this is just one of several examples:
Source: Hoover Institution Stanford University
Among civilians, foreigners and people in occupied territories were among the first to have identity cards, as (for example) in France in 1917 and the example below issued by the British authorities in Palestine in 1930.
Attrib: Par mickyx09 [CC BY 2.0 (https://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
In 1938, Nazi Germany introduced the Kennkarte for all civilians, also known as "Ausweis" (thanks to jwenting for this word), and this became a requirement in occupied territories.