Is the Apple rapid response message for my iPhone, real or fake?

As you can see by the support document you posted from, they are real.

Apple is now issuing time sensitive security patches directly without requiring a full iOS version update.

Also, there is no way to get fake system messages with updates like that. If it shows up in the Software Update section of the Settings app on your iPhone, it is 100% real.

Real, as explained in the support article that you were reading just before you posted.

That is a true update from Apple. You can safely install it.