Is it possible that every M has a name beginning with M?
Not always but almost:
For Bernard Lee's M it was hinted that his real name is like it's comic book counterpart Sir Miles Messervy.
While his name was hinted at throughout the novel series and finally revealed in The Man with the Golden Gun, in the film series only his first name, Miles, was revealed in The Spy Who Loved Me. - wikia
For Robert Brown case it's more tricky as it was never stated explicitly that he continuted role of Admiral Hargreavesin from The Spy Who Loved Me or continuing Bernard Lee's role. But if he playing as Hargreavesin then it suggest his demotion:
One thing to note is that in the pre-title credits to The Living Daylights, M's insignia suggests he is a Rear Admiral, which would mean if he was Hargreaves then he was demoted since Hargreaves' insignia in The Spy Who Loved Me suggests he is a Vice Admiral. - wikia
For Judi Dench version not her first name but her surname start with M, from dailymail.co.uk:
During the scene, Bond, played by Daniel Craig, is handed a box containing a Royal Doulton bulldog draped in a British flag, which is usually kept on M’s desk.
Zooming in on the box’s inscription, the words ‘From the Estate of Olivia Mansfield Bequeathed to James Bond’ are visible.
Even officials confirmed it:
Meg Simmons, archive director for Bond production company Eon, said: ‘We’ve searched around and as far as we can see this is the first and only time anyone’s ever revealed M’s real name.
‘It may not have been spoken, but if you were watching on a big enough screen it could have been visible, so we’d argue it’s now canon.’
And for Ralph Fiennes you already know that his name is Mallory.
Side-note: But remember in the original Fleming's novels there was only one M which was revealed to be named Vice Admiral Sir Miles Messervy KCMG, CB, DSO, OBE.