When Philip became a British citizen in 1947 (before his engagement to then-Princess Elizabeth was announced), he took the Mountbatten name officially.

As to WHY, that’s a little more difficult to say precisely as there may be some hidden (okay, not-so-hidden) family dynamics involved. After the fall of the Greek monarchy in 1922, Philip’s family essentially bounced around Europe to different relatives for awhile. From about 1930-1933, his father settled in Monte Carlo (which Philip may have viewed as pretty much abandoning the family), his mother was diagnosed with schizophrenia and hospitalized, and his older sisters married, breaking up tje family entirely. The Mountbatten side of the family stepped in and took care of Philip for the rest of his childhood, and helped him launch his career. His close relationship with his uncle Louis “Dickie” Mountbatten as a surrogate father is well documented, and Philip May have chosen the Mountbatten name to break completely with his biological father and honor Louis instead.

Setting aside personal issues of family loyalties, there may also be an incredibly simple, practical explanation. Mountbatten sounds more British, and as Philip was about to become a member of the royal family when he took the name, that made sense as a PR move. On a related note, possibly because Mountbatten sounds British AND specifically not German, unlike Glucksberg. In fact, the Mountbatten name had originally been Battenberg, and was changed by Phillip’s maternal grandfather Louis Battenberg/Mountbatten (the father of the more famous Louis Mountbatten who was also known as “Uncle Dickie,”) during WWI. Anti-German sentiment was again running high in the aftermath of WWII, so it would again make sense for Philip to choose a surname that sounded as “homegrown” as possible.

Meanwhile, Philip’s sisters all married German princes, and kept the German heritage in their names. They also had little to do with the Mountbatten side of the family to begin with, so there is less of a connection there either.

Prince Philip's issue is not Glucksburg-Windsor because he was not Philip Glucksburg (or rather, Philip Schleswig-Holstein-Sonderburg-Glücksburg) when he married Queen Elizabeth. He renounced his surname, foreign titles and Greek citizenship at the same time as naturalising to be a British subject on 28 February 1947. But that's a very brief answer.

On the 22nd of October 1947, Clement Attlee called for a motion in the House of Commons, to present "an humble address" to the King and Queen, congratulating them on the engagement of Princess Elizabeth to Prince Philip Mountbatten, who had announced their engagement during that summer's parliamentary recess.

The MPs were largely congratulatory, up until the controversial MP from West Fife, Willie Gacher of the Communist Party of Great Britain, objected.

I cannot forget that on the day that this engagement was announced, thousands of Greek citizens—Communists, Socialists and trade unionists—were thrown into the prison camps of the reactionary Royalist Government.

We are told that this young man has forsaken his family name and nationality, and has taken on another name and another nationality. But that is of little consequence, because I am quite certain that he has not forsaken the family politics. The "News Chronicle," with unctuous hypocrisy, informed us that as Philip Glucksburg he would have been viewed with suspicion, but as Philip Mountbatten he will be warmly welcomed in this country. That is what may be called flying in the face of Shakespeare.

This was after the name change (and Prince Philip wasn't trying to win over communists). To understand more, it's helpful to look at the background.

** Why the name change? **

Although Prince Philip changed his name to his mother's maiden name of Mountbatten, Princess Alice had never been called Mountbatten. Instead, she had been Her Serene Highness Princess Alice of Battenberg, until she married into the Greek Royal family in 1903.

Princess Alice's father, Prince Louis Alexander of Battenberg, had ascended to the rank of First Sea Lord - essentially head of the Royal Navy, reporting to The Admiralty. This was despite being a German born Prince.

Ultimately owing to pre-existing strained relations with Germany (which had been building against Kaiser Wilhelm II since the Kruger Telegram and the hilarious Daily Telegraph Affair) and then the outbreak of war and a tense mobilisation exercise, Battenberg sent a letter of resignation to the First Lord of the Admiralty, Winston Churchill in October 1914:

I have lately been driven to the painful conclusion that at this juncture my birth and parentage have the effect of impairing in some respects my usefulness on the Board of Admiralty. In these circumstances I feel it to be my duty as a loyal subject of his Majesty, to resign the office of First Sea Lord, hoping thereby to facilitate the task of the administration of the the great Service, to which I have devoted my life, and to ease the burden laid on H.M. Ministers.

By 1917, the Battenbergs made the decision to change their name to Mountbatten and relinquish all German titles. This was days before the 17th of July 1917 when King George V issued

A PROCLAMATION declaring that the Name of Windsor is to be borne by his Royal House and Family and Relinquishing the Use of All German Titles and Dignities.

Because yes, the House of Windsor had previously been known as the House of Saxe-Coburg & Gotha (and before Queen Victoria married Prince Albert, the House of Hanover). Essentially the British Royal family wanted to distance itself fully from Germany.

So that leads us back to Prince Philip Glücksburg. Having departed Greece during the creation of the Second Hellenic Republic (wherein George II of Greece was deposed) his branch of family wasn't very wealthy, but he spent a lot of time growing up in the UK, and fought in the British Royal Navy in WWII (to the point he was mentioned in dispatches). Not ostensibly significant ties to Germany, and he fought for the UK, so seemingly a smaller need to change names?

He was sixth in line for the throne of Greece at the time of him renouncing his titles. He was also somewhere in line for the Danish throne, albeit a lot more distantly (and he would have lost that line of succession by 1953, when the rules of succession there got changed). He was a pretty minor royal - but in theory if a bunch of people died then the King of Greece would be married to the Queen of England, and Kings always outrank Queens (hence why he was Prince Consort and not "King Consort").

It's specifically worth noting that King George II of Greece got to be King again, as Alexandros Zaimis, the President of Greece, had been ousted and then a fraudulent referendum was run by Georgios Kondylis, wherein 98% of the vote decided Greece wanted to do kings again (this was supposed to be so Georgios could run the country with the King as a puppet), also then the King decided to back authoritarians who modelled themselves after fascist Italy called the 4th of August Regieme, but then that fell apart and the British Foreign Office got involved and then Nazis had invaded, and then he got to be King again after the war after another fraudulent referendum. Also some of Prince Philip's sisters had married Nazis.

So it was likely both an ingratiating move with the British Royal and public to eschew his titles and name.

Sources:

Britain's Royal Families: The Complete Genealogy, Alison Weir, 2011

Hansard HC Deb 22 October 1947 vol 443 cc83-7

Admiralty Correspondence (1660-1976) in the National Archives, ADM 196/38. f. 16.