I can see your point. And I agree.
But I doubt the Cambridge will name the child Matilda or Jane, in order to avoid such a tricky situation.
Imagine, if they name the Baby Arthur, and he decides to reign as King Arthur II. He would be recognising the mythical King Arhtu as a true English Monarch.
In fact, Empress Matilda and Lady Jane Grey are listed among the English Monarchs, at the British Monarchy's website.
History of the Monarchy > The Normans > Stephen
History of the Monarchy > The Tudors > Jane
I doubt the Cambridges will name a daughter Jane simply because it's not a name that has much of a tradition within the family. It's appeared twice in the direct line - once as the name of a queen consort, Henry VIII's second wife, and once as a (disputed) queen regnant, Jane Grey. Matilda, however, has been rather popular and appears 8 times since the Conquest (making it tied with Eleanor for the 4th most popular female name). The Mountbatten-Windsors have a history of reintroducing older, pre-Hanoverian names to the direct family, so I personally wouldn't be surprised if Matilda was considered as a possibility.
Your comparison to King Arthur, however, is a bit of a stretch. For starters, the numbering system in the English tradition is based on the post-Conquest monarchs (there were 3 Edwards pre-Conquest, but Edward I was still numbered such), and as such on that basis alone any Arthur (or Alfred, Edmund, Edgar, or Harold for that matter) would be the first of that name. Even if we were to consider the pre-Conquest Wessex and Danish kings in the numbering, we would also have to consider whether or not they were Kings of England, or kings in a kingdom that would later become part of England. Alfred the Great was a king of Wessex, not England, and as such even allowing for pre-Conquest English monarchs in the numbering, any future King Alfred would be the first of that name.
Back to Arthur, though, there is also the fact that his mere existence is debated. There is no debate as to whether or not Matilda or Jane existed, there is a debate as to whether or not Matilda and Jane were actual monarchs owing to the conflicts surrounding their claims and the fact that they were both rather quickly disposed. Arthur is a legendary figure who is believed to have possibly existed 4 to 5 centuries prior to the Conquest and if he did exist was in all actuality a Welsh or Cornish monarch and not an English one - if he existed he ruled in a around the Anglo-Saxon invasion and his people were the ones defeated in said invasion.
Details, details!
I personally don't give a hoot about the annointing. Stephen had no right to be king and he should not have been. I choose to consider Matilda as Queen. I don't expect anyone else to agree with me on this point.
By that logic, Matilda's father, Henry I, had no right to be king and should not have been king, as he too usurped the throne. Stephen did have a claim to the throne - through his mother - and he took advantage of Henry I's lack of a male heir and his being in England at the time (Matilda was in Anjou when her father died).
You're wrong, though. Stephen did have a claim to the throne - his mother was the daughter of William I. Matilda had a better claim - her father was the son of William I and king in his own right - but that doesn't negate Stephen's claim.
You mightn't give a hoot about the anoiting but to the medieval mind, and even the more modern church mind that is a crucial sign of monarchy.
It is a major reason why Edward VIII was forced to abdicate before his coronation rather than after - once annoited getting rid of him would be much harder for the church to support.
In the medieval world being anointed was seen as a sign from God, through God's official representatives here on Earth of the right of the King - thus Stephen was the anointed King and Matilda's only way to remove him was through his death - anything less would have seen her and her kingdom excommunicated meaning the the church would tell the people she wasn't the monarch and that they didn't have to follow her lead.
It's arguable as to whether or not Edward VIII was forced to adbicate or merely pressured into it. I somehow doubt the church would have had too much of a problem with said abdication had he been annoited, given as they were pushing for the abdication in general over the idea of him marrying Wallis and having her become Queen.
The idea that Matilda may not have been a monarch doesn't necessarily stem entirely from her not having been anointed though. Edward V was never anointed and yet we still consider him to have been a monarch. The problem with Matilda is that at no point was her reign not disputed. Stephen usurped the throne immediately - as such she did not inherit. The point of time during which it's contested that she may have been a monarch is actually 6 years after the death of Henry I. In February 1141 she defeated and deposed Stephen, was welcomed in London by its citizens (which is often a mark of being the monarch), only to be deposed by Stephen in November. In my opinion, this is comparable to the second reign of Henry VI, who had first ruled from 1422-1461 before being deposed by Edward IV, then reconquered his throne in 1470 and then ruled again for a few months before being deposed by Edward again.
We recognize Henry's second reign because he was already an anointed King, but it's one that was contested at the time, was very short, and ended in him being usurped. Matilda was never anointed, but she was the legitimate heir who was usurped on her father's death, deposed her usurper in 1141 and was accepted by some of the people as monarch for a time before the civil war broke out once again and she was deposed. That she never had a coronation does not mean that she herself did not conquer the throne, however briefly. She's not the only monarch to not have been crowned - Edward VI, Jane, and Edward VIII weren't crowned, and in Scotland Margaret of Norway wasn't crowned, but only the female reigns are disputed.
