Q:

What personal identification documents does the British Queen have?

What documents does the Queen use to travel abroad and to identify her personally domestically?
Candidates include:

A UK identity card
An EU passport
A diplomatic passport of a sort
Any ID card replacement that may be used for personality identification. In various countries it may be driving license, military ID certificate.
Any royal certificate or diploma

etc.

A:

As a general rule, the Queen of the United Kingdom doesn't need any state-issued ID, because she is the state. She doesn't need her own permission to do things, or to verify to herself who she is.
Specifically, she has neither a passport nor a driver's license. (ref, ref) 
There is no such thing as an "EU Passport", just passports issued by EU member nations.  Since she is her own passport, she doesn't need one to enter a foreign country.  (And if they turned her away because of her lack of a physical passport, it would be a major diplomatic incident.)
She does not have a diploma of any sort, as she never graduated from a university (although she might hold honorary degrees - I didn't look).  That said, this is one kind of documentation it does make sense for her to possess, as it would be the university issuing it to her, not the state.  She just happens not to.
There is no "royal certificate" equivalent to a birth certificate, but she does have a marriage certificate, as documented in one of the books linked here.

One thing she does have is her various Coats of Arms (and Flags) for each country.  They are unique to her in her position as Monarch, and will be inherited by her successor upon her death (or abdication).  The government and official documentation (such as everyone else's passports) use a modified version, as do her descendents.  It is illegal to use them deceptively.

A:

Inside the front cover of my passport, the following statement appears;

Her Britannic Majesty's Secretary of State requests and requires, in the Name of Her Majesty, all those whom it may concern, to allow the bearer to pass freely without let or hindrance, and to afford the bearer such assistance and protection as may be necessary.

I would think that the Queen herself doesn't need her Secretary of State (which we know commonly as the Foreign Secretary) to ask on her behalf that she be let into a country.
All a passport is in essence, is a document from the government saying you are who you are, and you have some authority to leave the country. The UK Government is de jure the government that represents the monarch - the executive arm, if you like, of the sovereign.
In common law, she doesn't require any of this documentation because of the royal prerogative. This embodies the power of the monarch, and in the United Kingdom, the royal prerogative can only be changed using a special procedure.
To just take each item of your list in order though;

A UK identity card

There is no such thing since the scheme was scrapped in 2010.

A EU passport

There has never been such a thing as an EU passport, but she hasn't got a UK one either.

A diplomatic passport of a sort

No.

Any ID card replacement that may be used for personality identification, such as a driving licence.

No, and she doesn't need a licence to drive a car either, for the same reasons that she doesn't require a passport. She would effectively be issuing it to herself.

Any royal certificate or diploma

Erm, no.
What she does have, however, is a birth certificate. When she was born she obviously wasn't the queen (indeed she wasn't expected to become queen in the future as her uncle was the heir to the throne), so the government - through the local authority - issued a birth certificate. Monarchs also have death certificates, as when they die they cease to be the monarch.

A:

First of all, not even commoners are legally required to have a UK identity card, much less the Queen.
As for the EU passport, it simply doesn't exist, so she doesn't have one.
The official website of the Royal Family clearly states that the Queen doesn't have and doesn't need any kind of passport:

When travelling overseas, The Queen does not require a British passport. The cover of a British passport is in EU format, maroon in colour and features the Royal Arms. The first page contains another representation of the Arms, together with the following wording:
'Her Britannic Majesty's Secretary of State requests and requires in the name of Her Majesty all those whom it may concern to allow the bearer to pass freely without let or hindrance and to afford the bearer such assistance and protection as may be necessary.'
As a British passport is issued in the name of Her Majesty, it is unnecessary for The Queen to possess one. All other members of the Royal Family, including The Duke of Edinburgh and The Prince of Wales, have passports.
In realms (Commonwealth countries where The Queen is Sovereign), a similar formula is used, except that the request to all whom it may concern is made in the name of the realm's Governor-General, as The Queen's representative in that realm. In Canada, the request is made in the name of Her Majesty by the Minister of Foreign Affairs.

