Thanks to the incomparable Natalie Zemon Davis, the most famous (today) case of early identity theft is actually a 16th century French peasant! Martin Guerre (the actual one; set aside the movie for a moment) vanished from his southwestern French village in 1548, perhaps fleeing accusations of theft. He left behind a fairly comfortable life with his wife Bertrande.

With no record of his death, she was not permitted to remarry. Luckily for her, in 1556 Guerre returned! He and Bertrande had two children and took other members of his family to court to secure his/their fair share of the inheritance from his now-deceased father. Villagers raised doubts about Guerre's identity, of course, but his wife stuck by him. It was the real Guerre, to be sure.

Except--it wasn't. The relative who had lost that lawsuit, Guerre's uncle Pierre, did the detective work in tracking down the identity of the impostor, and secured some evidence for why he couldn't possibly be the 'real' Guerre. (A soldier related to him the story that Guerre had lost a leg fight in war.)

Still, fortune seemed to favor the impostor...until, at the appeal of his trial where Pierre and Bertrande's own reputations and futures were on the line as potential perjurers and false accusers, the actual Martin Guerre showed up, lack-of-leg and all.

Something that I see from time to time is information being used or stolen after someone has passed away. Even with death records, people still try to take advantage of it.

After Joan of Arc's execution, multiple Joans popped up in France. One of them in particular, Claude des Armoises, became quite well known: wined and dined and feted across the country. Some of this tracked with rumors that Joan had survived her execution, or that someone else had been burned in her place; some of it seems to have been more a hope/desire to believe against knowledge.

And beyond actual cases, it was certainly a concern, as the popular literature of the era reflects. For example, in one of the most famous iterations of the Arthurian saga, the 13th century Vulgate cycle, Arthur nearly burns Guinevere alive because he has been convinced her identical half-sister is actually her. Of course this is legend and magic is involved, but it still speaks to the danger of lookalikes and impersonation.

Slightly outside of the scope of your question, but I can tell you about a case of identity theft in the ancient world: possibly the first recorded case in Europe.

Sometime between 343 and 340 BC, a woman called Neaira (or Neaera) was brought to trial accused of passing herself off as an Athenian in order to illegally marry a man called Stephanos. The prosecution speech against her survives, as part of the Demosthenic corpus (though Demosthenes certainly didn't write it: it's usually attributed to the main speaker in the case, Apollodoros).

As part of the speech, Apollodoros mentions Neaira's (or Stephanos', or Neaira and Stephanos') daughter, "Phano, who used to be called Strybele". Exactly what Apollodoros meant by that remark has been debated by historians, but in 2009 David Noy proposed that the solution was "a case of Athenian identity theft".

Other suggestions to explain this have been that having multiple names was a sign of a prostitute (the famous prostitute Phryne's real name was in fact Mnesarete, for instance), which is possible, but simply speaking a woman's name implied that they were disreputable, so it's not clear whether suggesting that they had changed their name was any more shocking. Christopher Carey suggested that "Phano" was an especially inappropriate name for a respectable woman because it was similar to the verb 'phaino' ('to show'), and Classical Athenian women ought not be seen in public. We know of two Athenian citizen women called Phano from the fourth century, though, so this seems implausible.

The important point with identity theft in classical Athens is that it would have been really easy. Women were known by their relation to men, and it was entirely possible that for instance, there would be two women both known as X's sister, or X's daughter. There would have been witnesses to the dekate (the naming ceremony when a child was ten days old), so witnesses would have been able to testify that someone had a daughter -- but proving that a specific woman of about the right age was or was not the same one who had been named many years ago would have been basically impossible.

In this specific instance, Phano married, divorced (in scandalous circumstances, it being alleged that she wasn't the daughter of Stephanos), and then remarried Theogenes, the archon basileus (who was required to have a virgin bride). Her history came out, and it resulted in a second divorce. Noy suggests that Theogenes married her not because he hadn't heard about the scandal, but because she was passed off as a different daughter of Stephanos.

So, at least according to Noy, Phano might have stolen the identity of as many as two daughters of Stephanos who died as children. (Another case of identity, Demosthenes 39, is not quite a case of identity theft, but concerns two half-brothers who both lay claim to the same name).