Thanks to Autodidact's comment with a link to an informative, documented web page, the following points help in understanding Paul's ability to be accepted as a Roman citizen, assuming he did not carry any written documentation, as seems reasonable to me, which I posted in my question, but also noted in the link, "Romans did not carry around ID cards or documentation proving their Roman citizenship with them all the time."
First, Paul seem's likely to fall under the category of a Provinciales citizen, who "were people from the provinces who were under Roman control or influence, but who only had basic rights under international law (ius gentium)" (unless otherwise linked, all citations are from the link given above). This is based on the fact that he does testify that his citizenship was tied to his birth place of Tarsus (Act 21:39, 22:3), which was in the Roman province of Cilicia. People born in such (along with children of people who earned their freedom) "were born free citizens," as Paul asserted in Act 22:28. Apparently those "basic rights" included not being mishandled without trial, and I would suspect included some similar at least to the non optimo iure rights of the cives romani who were "full Roman citizens. They were subdivided into two classes: the non optimo iure, who had rights of property and marriage and the optimo iure, who also had the right to vote and hold office."
Second, cultural clues were important:
Language and clothing also played a role in determining if a person was a Roman citizen or not. An individual who spoke good Latin, who behaved and dressed in certain ways, displayed his status and Roman identity. Only Romans could wear the toga and it was strictly forbidden for non-citizens, foreigners, freedmen and slaves to wear it in Roman territories
In Paul's case, he had demonstrated to the commander previously that he spoke Greek (Act 21:37), though that was not obviously enough to itself prove he was a citizen.
Third, "Roman names were ... a sign of Roman citizenship." So Paul as Paul (his Roman name), rather than as Saul (his Jewish name), was a sign of his potential citizenship. Roman names
were a way to immediately identify the status of an individual, his clan and family. Traditional Roman names for male Roman citizens consisted of three parts (and not two as it is usually the case today): the praenomen, the nomen and the cognomen. The nomen was the name of his clan. The praenomen distinguished individual members of the clan from one another while the cognomen was the first name
Scripture does not explicitly state what Paul's three part name might have been, but it can be assumed, since he was a male citizen, he had one. It may hint at it in Php 3:5 (NKJV),
circumcised the eighth day, of the stock of Israel, of the tribe of Benjamin, a Hebrew of the Hebrews; concerning the law, a Pharisee;
as it could be his nomen (clan) was Hebraica (Hebrew) or Yisrael (Israel), his praenomen (family within the clan) Benjamin, and then his nomen (first name) Paulus. The more Hebrew related names would have been transliterated into Latin. But it may be he had some other designation than these. In any case, his full Roman name would have been a verbal indication of his citizenship, and could have been testified to in both Acts 16 and 22, even though Scripture did not record that testimony.
Fourth, the linked article mentions the five year census of tribal lists. Now Paul was born in Tarsus, but when Paul says (NKJV, bold added), "I am indeed a Jew, born in Tarsus of Cilicia, but brought up in this city at the feet of Gamaliel" (Act 22:3), which means for most of his life, he would have taken his census in Jerusalem, and that even was done mainly by verbal testimony.
1 So in the case of Acts 22, the commander could have followed up in checking Paul's verbal declaration by consulting census records of Paul in Jerusalem. This would not have been possible for the Acts 16 scenario.
But note that being a citizen (πολίτης) of Tarsus (declared to the commander in Acts 21:39) was not enough to indicate he was of Roman citizenship of that city (since the commander was planning to treat him as a non-Roman even after that, Act 22:24; the commander refers to his citizenship as πολιτεία, but I do not believe there is a distinction in the terms, but only if one is considered not just a citizen of a city, but also of the larger empire of Rome).
Fifth, regarding proof while travelling abroad, the article mentions "Romans could produce the grant of citizenship or their birth certificate," but in absence (or doubt of) that document, "witnesses could be called." And the article gives an example and expands on that saying (bold added),
what mattered most in the Roman world was a citizen's social network rather his legal status. A citizen's social network could impose consequences and provincial governors / foreign authorities had to beware of the consequences before mistreating any Roman citizen, especially an upper class Roman citizen. More than the documentary evidence, what mattered the most was the citizen's connections.
Earlier in the article it stated:
When in doubt, anyone could just ask around about a person's social standing and reputation. This made even more sense in a society where those who could read and write were the exception rather than the norm. In a small town, word of mouth was often the only way people had to prove their Roman citizenship.
So for the Acts 22 incident, Paul was plenty well known enough in that area to have witnesses confirm his citizenship status (should there be a doubt). In Acts 16, it could be that he would have also had a reliable set of witnesses in his association to Lydia (a woman of some substance it seems, from Thyatira, v.14) and his recent association directly to the jailer (vv.29-34), though certainly obtaining such corroboration in that location would not have been as easy as in Jerusalem. It seems, however, that the officials saw no need to corroborate the claim.
So it does seem, that even for census purposes, a simple verbal declaration, specifically including one's full tripartite Roman name, was enough to satisfy most Roman authorities that one should be treated as a citizen. That, coupled with possible language, clothing, and behavior clues, would help authorities assess the truthfulness of a verbal claim.
Should doubts arise, relationships to witnesses able to testify to knowledge of the person could be consulted as follow-up, or even, depending on one's location to a previous place of lengthy residency, census records from a prior census.
1 As the Wikipedia link notes, the census itself was often merely by verbal statement:
According to these laws, each citizen had to give an account of himself, of his family, and of his property upon oath, "declared from the heart." First he had to give his full name (praenomen, nomen, and cognomen) and that of his father, or if he were a Libertus ("freedman") that of his patron, and he was likewise obliged to state his age. He was then asked, "You, declaring from your heart, do you have a wife?" and if married he had to give the name of his wife, and likewise the number, names, and ages of his children, if any.
So no written documentation was necessarily needed or expected during the census, though obviously, if one had it, it may come in useful to speed things along.
Regarding location, even though Joseph returned to his family's ancestral home with Mary during the census (Luke 2:1-5), that does not mean the practice of travelling to an ancestral home was common or expected. The registration was to occur by a person going "to his own city" (Lk 2:3), which more likely indicates the city near which one currently resided (since it does not appear that Romans had to criss-cross the known world just to "register" themselves back where they or their clan were located). But ultimately, the census occurred where the censor (the one conducting the census) was located, and Luke simply records that during this time of census, Joseph was travelling away from Galilee, with Mary, and so chose to do his census at his ancestral home of Bethlehem.