What it is
Someone says something in a specific context. Later retellings detach it from that context, change the wording a little, or credit it to someone more famous. Each step adds authority and removes provenance. The end state is a quote everyone recognizes, nobody can source, and that says something slightly different from anything the credited person believed.
It can happen through careless repetition or deliberate editing. Only the record can tell you which.
How it shows up for me
The internet is full of laundered quotes, and so were my old notes. Now, before I quote someone here, I try to find where they actually said it. When I can’t, I say so, like the line about Rome’s roads in deferred maintenance.