The general rule
Birth certificates, social security cards, and driver's licenses
identify a person, but what happens if these are all lost?
Say a homeless person loses all of their documents in the shuffle,
what could they do to recover them? Even further, if this person has
no family or work colleagues who will vouch for them, is it possible
that their identity is lost forever?
Generally, these can be replaced.
For example, I was robbed at gunpoint a couple of years ago and the robbers took (among other things) my driver's license and Social Security card, which were never recovered (I did recover one prescription slip from a dumpster about a mile away that was wrapped around a syringe, because the robbers were also injection drug users.)
I went by myself without any ID to the DMV which had an online record of my driver's license containing my age, height, weight, sex, race, noted that I needed vision correction and also had my most recent driver's license photo and a fingerprint. I explained what happened without any corroboration, and they promptly issued me a new driver's license.
The same process would have applied had I had a state ID in lieu of a driver's license because I wasn't licensed to drive for some reason (e.g. if I was blind). The process in Texas would be very similar (I don't know if they have finger prints though).
My daughter had to do the same when she lost her driver's license while camping.
With the driver's license, I was able to go to the Social Security office and have them reissue a Social Security card.
One of my children's birth certificates was lost, and I could simply go to the Vital Statistics department with a name and date of birth and get a new one.
The replacement birth certificate and driver's license involved a modest fee (which would be pretty daunting for a homeless person), but the replacement Social Security card was free.
It's a pain in the neck to do this, and it took several hours to sort out (the time lost would not be a problem for a homeless person), but loss of my identity was not a serious possibility.
If you have (or had before you lost it) a photo ID such as a driver's license, or state ID, or student ID, or passport in a system, reconstructing your identity isn't that hard.
If you don't know who you are either.
If you have amnesia, so you don't know the information needed to recover your records, it can be much harder to work out a lack of any ID. This happens something on the order of several times a year. Sometimes it is resolved promptly when the person regains their memory or is tied to a recent outstanding missing persons report, or is identified after a local TV broadcast seeking input from the public. But, if the person is not local, no one filed a missing persons report, and the memory loss turns out to be permanent, it can take months or years to get it sorted out.
But, you will generally be assumed to be legally present in the U.S. until proven otherwise if you speak fluent English. Also, you can't easily be deported if no one can determine your nationality (including you), even if you don't speak fluent English.
For example, if the only language you spoke fluently was an Amazonian tribal language and no one could figure out this fact, it would be hard to deport you without evidence of your country of origin, which by assumption, does not exist in this scenario. Officials sometimes try to crowdsource recordings of someone speaking or writing in these situations to determine their place of origins, which can take many months and isn't always successful.
But, if your first language was Klingon, you would be quickly identified and not deported, because that language is widely recognized and a Google search would reveal that it has only two native speakers, both of whom are children who were born in the United States (I know their father as a casual acquaintance).
Citizenship, arrest and deportation
Can a person lose their citizenship in this manner? Could they be
arrested or even deported?
You cannot lose your citizenship in this manner, although it can be harder to prove your citizenship, if you need to do so.
You shouldn't be arrested for simply not having ID (although it does happen) if you aren't driving a car without a license. Also, lots of people without ID are arrested all of the time for other charges, and then refuse to provide anything but a false alias to the police. But, they are usually not deported unless there is some reason to suspect that they are not U.S. citizens.
A modest but significant number of people every year (on the order of dozens to a couple hundred) are arrested and deported in circumstances like these and the system can put someone in that situation in a very Catch-22 situation for which there a few if any legal remedies after the fact by way of compensation.
Deportations Of U.S. citizens with strong foreign ties
The hard cases are usually those when you have someone who is a child who has never had a photo ID, or is an adult who never had a photo ID in the U.S., especially if that person is someone who has lived much of their life abroad despite being born in the U.S., or is most fluent in a non-English language because their parents spoke that language when they were growing up, or was born abroad and naturalized as a citizen later (often as a relative of the primary person who earned the right to citizenship by taking citizenship tests).
Establishing that you are the same person as the one in a birth certificate or naturalization certificate is not always easy. Naturalizations often aren't accurately cross-referenced with immigration records. And, the track record of ICE agents meaningfully following up on claims of citizenship is very poor.
Identity assumption cases
Another much less common hard case that still happens sometimes, involves a situation where you are born in the U.S. or to a U.S. citizen, giving you U.S. citizenship, and you have a birth certificate, but you live abroad for a long time, and in the meantime, someone similar in age, sex and race to you has assumed your identity to claim citizenship status.
Then, there is someone else with a long paper trail that supports your identity actually belonging to them including photo IDs and maybe even a passport, and you have only your birth certificate that someone else claims is theirs with doesn't have a lot of provable biometric features other than your parent's names, which may be hard to use to establish that you are the true person corresponding to the birth certificate if they are deceased, and may require DNA testing or testimony from them even if they are not deceased.
Persons declared dead
A third class of people who have a thorny time re-establishing their identities are people who went missing, were declared dead legally by a court as a result, and then resurface. This can pose a problem even after they undeniably establish their identity due to rules relating to the finality of court orders.
A fictional example of this is the title character in the live action TV series the Iron Fist in the Marvel Comics Universe. But, there are also real life examples, such as a man in Ohio a number of years ago who was in a similar situation (except for the fact that he didn't have superpowers, was middle aged, and wasn't an heir to a billion dollar fortune).
These cases are very rare. There is probably less than one per year in the entire United States.