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. 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, and this also became a requirement in occupied territories.
Kennkarte, issued 1940