That's a felony in many places.
I ask the office clerk: "Hi, I'm here to retreive the documents for X". He checks my ID and gives me the documents without asking for written permission from the other X.
It's deception by omission, but there is no fraud. I was legally allowed to do this. It's also a win for everyone because it avoids complications.
You could easily be a paw in an identity theft play like that.
I suspect it may fall afoul of a law about making false statements to the government, but that's distinct from fraud.
Impersonating someone at their behest as a favour is unlikely to be dishonest.
Key note because in case someone decides to go bad faith here, I think the gp comments use case of fraud is a positive thing (if a bit dangerous). Redefining a term just because you don't like the pejorative implications is not a positive thing, though.
It's pretty simple: they are deceiving the other party that the document was handed to the right person, the consequences of which range from 'meh' to spending 6 years behind bars.
Let's say this was a summons for a court case. Then the judge would believe the papers were served properly, when in fact they were never served. Permission doesn't enter into it here, it is wilful deception by two parties of a third. The same name should not normally be enough to get away with this but assuming the story is real (there are many reasons to doubt this, such as the ID matching the name, but nothing else) it all depends on who does the checking. In some cases you might not even go home without a small detour in case you are found out (for instance, because the person handing the document over is familiar with the real recipient).
So, jurisdiction matters, what that document was matters, whether the permission was granted in writing (procuration), who the counterparty was, what the value of the document was, whether the parties lived in the same state/province or country and whether or not the permission was communicated to the person passing the document to the wrong recipient (probably not) and lots of other factors besides, such as the person giving permission still evading some legal obligation, which - conveniently - the story doesn't relate (and which hinges on what that document was).
And that's before we get into the wilder options of the second person being social engineered because after all, the only way they could be sure that they were acting on behalf of the real recipient would be to check their identity, in person, and with someone who is able to do that in a way that is legally binding.
Where's the crime?