What's with the secrecy of Judi Dench's M's identity?

There is a real world analogous to this situation. Dench's M is based on Stella Rimington, the first woman director general of MI5. Further, she was the first DG publicly named and subsequently photographed (by the paparazzi). If we look at it through the lens of Dench is modelled after Rimington, then it would make sense that during Bond's tenure she went from being completely obscured from the public eye to having to battle in public for the secret service.

My real guess is far more mundane: Neal Purvis (screenwriter for Casino Royale, Skyfall, and Spectre) simply forgot.

Both M's true identity and her identity as M are not top secret, but the connection between them is to be kept secret.

That's the point of their conversation, M stops Bond exactly where he would make the link (in case anyone was listening in on their conversation).

Think of it this way: both Batman and Bruce Wayne are publically known people The only secret that is kept is the connection, that Bruce Wayne is Batman, i.e. the link between the two otherwise publically known people.

And yet in "Skyfall" we see her appearing in an inquiry where her name - even if not uttered on screen even once - must be well known to everyone present, and there are various ministers (including Gareth Mallory, played by Ralph Fiennes), and even some members of the press with cameras and IDs, which contradicts the secrecy of M's identity.

There is actually a very recent real life example here.

During Michael Cohen's trial, Donald Trump was referred to as "Individual 1" to prevent leaking his identity. They did this even though it didn't actually keep Trump unidentifiable. The trial ended up exploring the fact that "Individual 1 won the 2016 presidential elections", and the winner of those elections is obviously Donald Trump. However, if the presidential elections had not been the subject of the trial, then this would never have been established, and Trump's identity would have been (reasonably) concealed from the trial itself.

The same applies to M here. Obviously, some people know M's identity. If no one knows something (even if it's top secret), then the knowledge (and the secret) doesn't exist.

an inquiry where her name - even if not uttered on screen even once - must be well known to everyone present

While there may be some who know M's name, that doesn't mean that M's name was made public during the trial. Maybe she was simply addressed as "individual 1" during the trial, but the script simply glossed over that.

You also need to consider why the secrecy is used to begin with. It's intended for operational security.

Let's say it's publically announced who is Head of the Secret Service:

mzywiol is the new head of the Secret Service

In the meanwhile, an enemy spy gets their hands on some of the secret service's documents:

008 was injured during operation StackExchange. They are recovering from their wounds in site zulu. M will debrief them.

There is nothing that establishes the public person (mzywiol) as a particular person in the leaked document. Maybe you're 008. Maybe you're M. Who knows?

Even if you know every person who works for the secret service, but not their codename, then that document doesn't tell you anything. Who is injured? Who is meeting them? Where are they meeting them? ALl important information is obscured behind codenames.

As there are more than 26 people in service of the Secret Service, you can't just assume that everyone gets their initial as their codename.

Apart from mentioned similarity between M and Stella Rimington "Casino Royale" reset the Bond time. It should be the first movie that begin whole series. It kinda create whole new universe for Bonds.
In "Quantum of Solace" you don't hear "My name is Bond, James Bond".
Now, with the "Skyfall" we see that new changes. Where spies and their bosses are no longer shadowy figures. Where Bond and MI6 need to answer questions.
In "Spectre" we see that the whole idea of MI6 and spies like Bond is questioned by government.

So it's a design that span on whole new series that Bond works and MI6 are no longer jolly blip rides and witty remarks. But a public affair that have impact on real life situation and if you blow enough space stations the space will hit you back.

I believe Judi Dench's (M) character name is actually Emma. Mentioned in the the movie Skyfall in the Bond family home by the caretaker shortly after she and Bond arrive. It is almost at the end of the film a scene or 2 prior where she dies in the chapel.

I believe her name was also mentioned by her husband groggily as she got out bed to check her laptop that Bond was accessing remotely from the Bahamas. The movie was either Casino Royale or Quantum of Solace. I can't remember which though.