Well I can answer for 17th century Britain. You are right that others vouching for you would be a large part of that, and particularly important in the 17th century would be having higher-ranking connections or patrons vouch for you. There is one story from Renaissance Italy, I think it was in the Decameron but I can't seem to find it, where a man is tricked into believing he is someone else because while he is in debtor's prison for the fictional person's debts he calls out to patrons of his as they pass and they ignore him. Your membership in a family and your social network were extremely important historically, and having respectable connections like a wealthy relative or patron would be key to establishing a disputed identity. We can also look at the 16th century French story of Martin Guerre where it was the acceptance of the family that led it to take so long before it was challenged. And the law required the suit to be in the wife name, and the wife's testimony was considered key. Personal connections are important here as well.

Another aspect though, is in 17th century Britain there were also fairly well-kept parish records for births, deaths and marriages. So certain details about their age and history were recorded, which could be compared to the person. Do they look the right age? Do they know their own birthday, or when they were married? Where are their children, do they know the names of their children? If they are impersonating someone who died ,they can check the funeral and burial section of the records.

People might also inquire about ancestry, especially with more well-born families. If they can't cite their ancestors, they might not be considered legitimate. Not only to prove a specific claim of identity, but something people were generally expected to know, again especially if they claim to be high-born. Even if you claim a unique title and not to be an existing person, if you were claiming nobility you would be expected to know how that title came about and from where you derive your claim to it. And of course, if you invent a title you would need to make convincing forgeries of letters patent for it, in case someone asks for them (or have a sufficient explanation for why they cannot be found, perhaps bolstered by other forms of records attesting to the history of your title).

If you were a time traveller going by your modern name or a version of it trying to make people think you are from 17th century Britain, it would also be useful to have more financial papers. Proof of owning a house for some time, or tenancy records would make a convincing case that you are an existing honest person and not an imposter or sneaky foreigner of some kind. If they have an Australian accent in 17th century Britain that might be problematic though. Modern Australian is not totally different from historical English, in some ways it is probably closer to historical pronunciation than modern British RP, but it is distinctive and includes some Irish influence (they could maybe claim they have mixed Anglo-Irish heritage?). Given it is also influenced by Cockney, people might think they have some heritage in the Surrey river Thames communities, some where like Putney, although it would not be quite the same. So they would certainly not make a convincing northerner or someone from the midlands. And the biggest problem is that modern Australian is generally non-rhotic, like modern RP, while historical English was usually rhotic until more recently. And they would need to brush up on their historical vocabulary and grammar, its not greatly different from modern English but there is a certain finesse to speaking it in a convincing natural way. If they don't know how to use thee/thou/thy/thine pronouns they might draw attention to themselves, although this was also the time period where you/your/yours, which were seen as more classy, were gradually replacing thee/thou/thy/thine.