Was England called England in the days of King Arthur?

The name, "England", is etymologically, Anglo-Saxon; that is to say, it originated with the arrival of the Angles tribe who migrated from Central Germany en route to the British isles 1500 years ago during the immediate aftermath of the Fall of the (Western) Roman Empire.

The word, "England", loosely translates as, "The land of the Angles". And in many foreign languages, a letter or a vowel is replaced by another vowel, so that the "A" in "Angles", is replaced or updated with an "E". Hence, "Engles", became, "the land of the Engles" or "England".

King Arthur would have probably referred to his homeland by the centuries old Roman name, "Britannia". The name, "England", would have been foreign sounding and Germanic in origin. King Arthur was, in all likelihood, of Celtic descent and probably more accustomed to Celtic and Latin sounding languages.

As another (good) answer went into, the Etymology of "England" goes back to the Germanic invasions that happened during the time of Badon Hill (the historical setting for the initial tales of King Arthur, roughly 500AD).

There was a recorded sermon discussing the recent history of the island written not long after then, that mentions the battle: Gildas' De Excidio et Conquestu Britanniae. That roughly translates from Latin as "The Ruin and Conquest of Britain. So our documented evidence is that locals of letters at the time referred to their island as Britain.

Another popular previous name was Albion (Ἀλουΐων in Greek), but that name appears to have gone out of fashion sometime after the first century.

Due to its etymology, most likely the term "England" initially only referred to kingdoms (lands) run by Old-English speakers. In a way, this is still the proper usage, since technically England is a country (separate from Scotland and Wales), and the island is Britain.

The first appearance of the phrase was that Arthur took the sword from the anvil and the stone. As others have pointed out, the phrase makes more sense if you view it as a miscopied version of took the sword from the Angle and the Saxon. Example source:

The unusual notion of a sword being stuck in an anvil on top of a stone might be accounted for by an early mistranslation or mix-up of words. Arthur is said to have successfully fought against the invading Anglo-Saxons, originally two separate tribes— the Angles and Saxons— from northern Germany. The Latin word for a rock or large fragment of stone is saxum, a word that sounds very similar to “Saxon.” This, together with the similarity of the name “Angle” and the word “anvil,” might explain how the unusual motif originated. If the legend held that Arthur had “drawn the sword” – in other word, “taken the fight” – from the Angles and Saxons, then at some point during the turmoil following the collapse of the Roman Empire in the 400s, and the subsequent lack of historical records, oral accounts may have become confused. By Robert de Boron’s time, an account of Arthur seizing the initiative from the Angles and Saxons might well have evolved into the story of him drawing a sword from an anvil and stone.

One theory has the original Arthur as a Welsh king named Owain Ddantgwyn (also spelled Ddanwyn) whose nickname was the Bear. Of course, Bear in Welsh is arth and in Latin is ursus. Put those together and you get Arthursus or Arthur. By that time, the Roman empire had given up on the British Isles, but many Romans had settled there. So it is sensible to think of a leader who tried to appeal both to long term native Welsh and new immigrant Romans by taking a name in both languages.

This is also consistent with the view of Arthur as a High King who united Britain (under whatever name). Of course, the "High King" had responsibilities more consistent with a more modern Duke. The other kings were more like barons, earls, and counts.

As you guessed, it is ridiculous to think of Arthur, the native Welshman who was resisting incursion from the Angles, as the king who unites the land of the Angles. He united against the Angles. If Arthur had been successful in the long term, there wouldn't have been an Angle-land (England). If Arthur had been less successful in the short term, it might have started earlier. England as England formed in the tenth century. Prior to that, there was Anglia. Arthur was the fifth and sixth century, after the time of Roman Britain.

The King Arthur of history is basically the Arthur mentioned in the Historia Brittonum written about 830 AD and in the Annales Cambriae written about 975. The HB puts the Twelve Battles of Arthur some time between about 450 and 550 AD, while the AC dates Arthur's victory at Badon in a year corresponding to 516 to 518 and the battle of Camlan where Arthur died in a year corresponding to 537 to 539.

Thus they were written about 292 to 438 years after Arthur allegedly died and it is possible that legendary elements have been added to those accounts.

In late Roman times most of Roman Britain was divided into city states called civitates based on former British tribes. Each city state had elected magistrates and a council and was divided into districts called pagi. Late Roman Britain had four or five provinces whose governors used various titles and supervised the governments of the civitates. The governors were supervised by the Vicar who ruled the diocese of Britain - government diocese, not a religious one.

In Cornwall, in Wales, and between the two walls in the North were British tribes that were not very assimilated to Roman culture and may still have had kings, who no doubt lived more or less peacefully with the local Roman military units and acknowledged the authority of the Emperor just as a number of 19th century Indian tribes lived in peaceful coexistence with local US military units and acknowledged the authority of the Grandfather in Washington.

Beyond the Antonine Wall were the Picts, who may not have been culturally British and were often enemies of Roman Britain.

The Diocese of Britain was part of the Praetorian Prefecture of the Gauls, which was part of the Western Roman Empire.

Post Roman Britain may have continued the Roman system of government and also reverted to Celtic forms of government. Leading men may have made themselves kings of each city state, and each pagus may have had a sub king, thus making maybe more than a hundred sub kings in Britain. So Arthur, who led the kings in battle, was probably at least a sub king in rank. Each province may have been ruled by a second level king, and the whole diocese may have have been ruled by a third level king.

Or city states may have continued to elect magistrates, and governors could have been chosen for each province, and a vicar to rule all the diocese. And maybe the vicar claimed to be loyal to the Roman prefect of Gaul and to the emperor. Or maybe the ruler of Britain claimed to be an emperor, a sort of north western Roman Emperor.

Or maybe the rulers of Britain used both Celtic and Roman titles simultaneously, the ruler of a city state claiming to be both a king and magistrate, and the ruler of a province claiming to be both a second level king and a governor, and so on.

Because there were several provinces in late Roman Britain, Britain was referred to as the Britains politically. Thus the vicar used the title of Vicar of the Britains. And so presumably in Arthur's era the overlord of all post Roman Britain would claim to be king, or vicar, or emperor, or whatever, of the Britains, not of Britain, and not of the Britons, though the title of King of the Britons is used in medieval sources.

The story of the sword in the stone is first mentioned by Robert de Boron in Merlin around 1200 AD. That is about 700 years after the time of Arthur.

https://en.wikipedia.org/wiki/Excalibur#Excalibur_and_the_Sword_in_the_Stone1

And so the Sword in the Stone is considered fictional.

As Wikipedia notes, "King Arthur is a legendary British leader who, according to medieval histories and romances, led the defence of Britain against Saxon invaders in the late 5th and early 6th centuries AD"

He was† therefore a Briton or Romano-British, not a Saxon, Angle, Jute or related tribe. This means that his native language would have been a Celtic language and that he lived at a time long before there was any notion of a political or geographic entity that became England hundred of years later.

Any Romano-British leader, of the sort the legend of Arthur is based on, seems unlikely to have a name for a southern subset of the many kingdoms in Britain at that time. If he did, he is unlikely to have named it after his enemies or to have used his enemies names for the lands in support of his claim to leadership.

Usually it is King Alfred, in the late 9th century, who is credited with the notion of combining the Kingdoms of Wessex, Mercia, Anglia, Northumberland etc into a single nation named (something not completely unlike) "England". Notions of swords in stones granting kingship of "England" must be later.

Footnote

† As suggested in a comment below, Arthur is a legendary figure. Legendary means, not merely famous, but nonhistorical and unverifiable. There were British or Romano British leaders who resisted the Saxon invasion of their lands. These leaders apparently did occasionally win a significant battle, though they were eventually largely defeated. A few of the exploits later attached to the name Arthur would have come from the actual exploits of various real British or Romano-British leaders. So instead of writing "[Arthur] was ...", it would have been more accurate for me to write "Arthur would have been" or "The real people, on whose actions some of the legend of Arthur was most likely based, were ..."