I'm specifically interested in the early 20th century in America. When did it become commonplace to need to carry a photo ID? How did you prove your identity without one?
I'm currently writing a dissertation that has a lot to do with identity in the later Middle Ages. It was actually quite difficult to identify someone; or, at least, it was easy to hide or skew your identity. I'll try to elaborate from some of the evidence I've encountered.
Identities, then as now, were multiplex and complicated. Thus you might answer the question "Who are you?" differently depending on the situation: are you at church? Are you at your parents' house? Are you abroad? Are you standing in front of a customs agent? Your "proof" of identity would also change based on these different situations.
In the middle ages, before finger prints and photo ID's, your identity was largely based on your community of residence and your religious affiliation. However, religious affiliation was encoded in layers of legal code as well. More practically, it was also based on physical appearance, language spoken, and cultural habits as well.
For some examples, I'll use some piracy cases I've been working with.
In one case, the King of Aragon sought to gain repayment for losses due to a pirate attack. He launched an investigation to figure out who the attackers were so he could demand repayment from their sovereign. Right off the bat it's important to realize they didn't know who the attackers were. The court investigation began interviewing people who were present. The witnesses gave as much information as they could: the captain of the ship was a "young man"; he had a beard; he seemed to be speaking Genoese, but then he also spoke some Sard as well. Some of the crew members were speaking Provenzal; others "appeared to be Genoese". After nearly 50 pages of witness testimony, the King ended up writing identical letters to both Genoa and the King of Naples, demanding the same repayment from both. In short, they still couldn't identify the pirates.
In another case--and this one is my favorite-- some Jewish merchants were hijacked by the Valencian Admiral on their way from Mallorca to Barcelona; since Mallorca was friendly with Barcelona, this shouldn't have happened. The Jews complained to the King based on this. But the Admiral responded that they weren't Mallorcan Jews... they were Moroccan Jews! The Jewish merchants denied this vehemently, and the king of Mallorca launched an investigation into their identities, finding that they had "homes, wives, children, and hearths in Mallorca". This, for the King of Mallorca, was about as solid an identity as one could have back then. But it wasn't enough for the King of Aragon, who still refused to repay the Jews.
tl;dr: your identity only mattered when authorities needed it to matter, and aspects of it could change and be manipulated from situation to situation.
Some of the answers for the questions here may be of interest to you http://www.reddit.com/r/AskHistorians/wiki/dailylife#wiki_proof_of_identity
Specifically this http://www.reddit.com/r/AskHistorians/comments/y27aq/how_were_people_identified_prior_to_photography/ (direct comment link) I think.
An interesting case related to this is the Return of Martin Guerre, by Natalie Zemon Davis -- a historical account of a 16th century French man who stole another's identity, property, and wife through extensive knowledge of the original man's habits and family, and through superficial similarities in appearance. The original man left to join the army in Spain and only returned at the end of the impostor's court case.
In Casanova's memoirs he rarely travels to a new place to live without assurance that either some third party has written ahead to an employer or sponsor saying that Casanova should be expected, or else given a similar letter of introduction either with a seal from a well known person or a definite friend of whomever he was going to meet. There are also several instances where he moves somewhere because he knows he had previously met someone who had lived there and on arrival he immediately seeks those people out. These people almost invariably introduce him to their friends either at businesses meetings or parties or coffee houses (which were a new cool thing at that time) and Casanova put a very high value on these opportunities to meet others.
In his early life he was employed as some minor functionary of the church (I forget his exact role) so maintaining his identity was important both to secure his income and access to the rich upper classes. At a few points in the book it seems like most of his income comes from gifts from very wealthy friends that he would pawn (always to a Jew).
This isn't your exact timeline -- most of his young life was spent in post Renaissance Italy, but it's the clearest example I have for something similar.