I’ve read a few things which talked about folks in the 1920’s or so, just packing up and disappearing. The implication is they travelled someplace and started a new life under a new name, but with no explaination as to how they did it. This was the pre-Social Security era, so they wouldn’t have to get an SSN until later on (and how did they do that without a birth certificate?), but surely their must have been some other means of identification that folks carried around with them at that point in time. What was it? Not everybody drove, so a driver’s license wouldn’t be the universal form of ID that it is now. Anybody know?
Birth records and land ownership records were kept. Geneology sites would have a lot of info. reagarding types of records kept on people.
Remember - there was no easy way to duplicate documents in that time - typewriters were still new, and slow.
I doubt that anyone would raise suspicion by coming into town without a copy of her/his birth certificate and introducing herself/himself as “brandshinynewname”.
County records were kept (more or less) for:
Births
Marriages
Deaths
Real property ownership
Voting
The idea of always carrying a state-issued ID is quite new in the US, and is (rightly, IMO) regarded with suspicion by some.
All you had to do was move to someplace new and start using a new name. There was no reason why anyone would doubt you or ask for ID.
Changing ID was very easy at one time. In the mid 70s you could still walk into any Social Security office and walk out with a card.
You could run up a debt and leave town, start over.
A birth certificate could be as easy as a handwritten “Joe Blow, 8 lbs. 10 oz., born to Jack and Jane Blow on March 14, 1894 in Hill County, Montana.” (Remember, someone born in Montana in the 19th Century probably would have been born on a homestead instead of a town.) and signed by a physician. If the signature is illegible, well, you all know how prescriptions look. It adds an air of authority in addition to foiling those trying to check up on someone. 
And, hell, if the county courthouse burned down, of course you wouldn’t have any records.
But you probably wouldn’t have even needed that much. Documentation was scarce until the vast urbanization and mechanization of the 1950s, and even then it was pretty easy to create a new life. I’m sure there are still Americans (de facto if not de jure) who have no form of actual ID still today, but try to get a job like that.
Genealogist here. The great majority of people born in the U.S. before the late 1880s did not have their birth registered in any civil registry at the time of the birth. After the Social Security Act was passed in 1935, and the first Social Security numbers were issued in 1936, a large percentage of the population was unable to prove their date of birth for retirement benefit purposes. Thus you will see a flurry of delayed birth registrations in county records circa 1941, for events that took place in the 19th century.
The main reason that the population schedules of the 1880-1930 U.S. censuses were indexed in the late 1930s and early 1940s was to help people prove their ages for Social Security purposes. If you could identify where you were living in the census taken nearest after your birth, the Census Bureau would look you up and make a certified transcript, which you could then use as part of the evidence needed to file a delayed birth certificate (other evidence might include baptismal records and affidavits from people who had known you since birth).
As for applying for a Social Security number, you could fabricate most of the information on your application, including your date of birth, and get away with it. The Social Security Administration asked for verification of your info only at the time you later applied for age-based retirement benefits. If you never filed for Social Security benefits, no one would be the wiser that you lied. It wasn’t until much later (1980s?) that the SSA asked for verification from the start.
I am currently helping an elderly gentleman find out what become of his father, who abandoned his family in the early 1920s and was never heard from again. On his marriage license (1917), his draft registration (1917), and his son’s birth certificate (1920), he lied about his name, his age, his place of birth, and his parents’ names. No one bothered to verify, apparently. We assume that after he deserted his wife and child, he lived under an assumed name (or reverted to his real name), as we cannot find him in an all-name index to the 1930 U.S. census.
Until the 1960s, the federal government discouraged the use of the Social Security number as a national identifier. But then in 1961, the Social Security number became your taxpayer I.D. number, and in 1967 it became the tag number for military personnel. My guess is it had to do with the introduction of computerized database files.