That's how language works. Eavesdropping once referred to water dropping off the eaves of a roof. There is mostly no "log" that "logging in" writes to. The menu itself has nothing to do with food or eating. Is the meaning clear to contemporary readers or listeners? Yes.
This use of "logging" comes from the "logbook", a book recording what happened on a ship, which was called that because a literal log (i.e. part of a tree) attached to a rope was used to measure the ship's speed.
An underrated fallacy in general. (Of course I'm just giving a common name to something that was identified, not accusing anyone of having the etymological fallacy in this conversation.)
> There is mostly no "log" that "logging in" writes to.
Um, there is, actually.
On the etymological front, "log" was originally an actual wooden log, dragged behind a ship. It then became the book in which measurements of the log's speed were entered.
The UCLA Computer Club had a log -- a book with dated entries. Someone augmented that with a wooden log ... an intricate culture developed around this pairing. Traces of this are preserved: https://americanhistory.si.edu/collections/object/nmah_69051...
(That drawing is a pretty good representation of the actual log. And there's another memorialized pun: MUS stood for machine unit second, which was the computing center measurement unit -- the Computer Club had a quota for submitted jobs [card decks].)
