King Henry V of England introduced a passport in the Safe Conducts act of 1414. http://www.theguardian.com/travel/2006/nov/17/travelnews
Photo IDs became more of a thing during he 20th century, when photography became more widespread. Before the First World War, most people didn't have any sort of identification. It wasn't necessary.
I'm curious about the thing you see in movies, when the soldiers say, "Papers please." What were on these 'papers'?
Enslaved men and women in 18th and 19th-century America carried passes when they had to travel away from the farm or plantation. These would need to be presented to patrollers, or any white person who asked, on demand. Sometimes these passes referred to the specific errand, but often these were passes that made reference to a regular visit. Many people had spouses or relatives on neighboring plantations, and they would visit every Sunday. In cases where the travel was very routine, within the neighborhood, and over familiar roads or rivers, the pass would not be necessary.
Free blacks in the South needed to have their "free papers" with them at all times. Even more so than slaves, carrying these papers was of paramount importance. Otherwise they might be suspected of being runaway slaves, questioned under torture, thrown in jail, and possibly illegally sold into slavery. In Charleston, South Carolina, free blacks wore badges depicting the liberty cap with the word, FREE.
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Speaking for the nineteenth century: IDs were not common or required. Others here are indicating that IDs were not necessary, this is, I think, a bit misleading. It wasn't that IDs weren't needed, there are several cases were IDs would have been very helpful, but I think it was more the enormity of the task of identifying and cataloging every single citizen that made it seem impossible.
One area where IDs would have been incredibly helpful would have been in criminal enforcement and the justice system. A constant problem was that criminals could give any name they wanted when arrested and so there was no way to determine if someone was a repeat offender or wanted somewhere else for another crime.
To solve this, a very clever man name Bertillon in Paris devised a system for identifying criminals. He invented the first mug shot and used a complicated system of body measurements to ID these individuals. Here's a write up on the Bertillon system and its adoption in the US.
Later this would be replaced by the finger-printing system.
Still- the average person would not be carrying anything that might identify them. In Ken Burn's Civil War they mention the soldiers sewing their name on the insides of their coats, but this is obviously a special circumstance.
If a dead body was found, no one would expect to be able to identify it, but people could identify the corpse if they had a friend or family member who was missing.
Sources: Cincinnati Enquirer, Louisville Courier-Journal, New York Times.