This is not true. There really wasn't "competition" for settling Iceland or Greenland (a term that appears to me to have problematic colonialist overtones, as Greenland would have already been inhabited by the Norse arrival in Greenland). So, in this answer, I'm gonna talk about whether or not anyone else in Europe actually knew about these far northern islands, what the stated reasons are for the naming of the two islands are, and why they are still called that today.

Did anyone know about Iceland

This is not actually as easy of a question as it seems. Since the 300s BCE various Greek and Roman authors talked about a place called "Thule". One of the most famous, and influential, comes from Pytheas of Massilia (Marseille), known today mostly through the accounts of Strabo and Pliny the Elder. Strabo is highly critical of Pytheas, saying that "through him many have been mislead." However, concerning Thule, Strabo claims that 1) Pytheas says that there the land, sea and sky merge into a hybrid such that a man can neither walk nor sail on it and 2) it is the farthest north of all lands, being six days' sail north of Britain. Pliny the Elder supplements this citation by Pytheas with two more - that in Thule, there is no night around midsummer and no day around midwinter, and that one day's sail past Thule, there is a sea of ice. (information taken from Ian Whitaker, "The Problem of Pytheas' Thule.")

Lots of different options have been proposed for where this is, and in fact it's likely that Pytheas and others who use the term mean different islands in the North Atlantic. However, it is distinguished from other islands like the Orkneys or Shetlands by Isidore of Seville and the Venerable Bede, before being attached to Iceland. Medieval Icelanders appears to accepted that their home was Thule - the Sturlubok rendition of Landnámabók, the Book of Settlements, starts by identifying Iceland as Thule:

Í aldarfarsbók þeirri, er Beda prestur heilagur gerði, er getið eylands þess er Thile heitir og á bókum er sagt, að liggi sex dægra sigling í norður frá Bretlandi; þar sagði hann eigi koma dag á vetur og eigi nótt á sumar, þá er dagur er sem lengstur. Til þess ætla vitrir menn það haft, að Ísland sé Thile kallað

In his book of ages, which the holy priest Bede made, there is this island which is called Thule and in the book it is said, that it lays 6 days sail north of Britain; there he said that day does not come in the winter and night does not in the summer, then when day is the longest. To this wise men have supposed this, that Iceland called itself Thule.

So, we definitely see some Classical awareness of the North Atlantic, and this gets handed down the Middle Ages. However, in terms of actually setting foot on the island, only one group can plausibly be said to do that: the papar, Irish monks that I've talked about before here. They're Irish monks identified in the twelfth century Íslendingabók, and supported by some archaeological evidence, but they're hardly a colonizing force.

In terms of reaching Greenland, though - there's really no evidence that any European peoples reached Greenland before the Norse did.

The Naming

So, what are stated reasons for naming?

ICELAND: In Landnámabók, the short version is that a man named Hrafna-Floki Vilgarðarson sailed to Iceland on the report of a Swede named Garðar Svavarsson, who supposedly landed in modern Húsavík. Floki sailed around Snæfellsnes and landed in the Westfjords. When he wintered there, all his cattle died, and the mountains all around were covered in ice.

Þá gekk Flóki upp á fjall eitt hátt og sá norður yfir fjöllin fjörð fullan af hafísum; því kölluðu þeir landið Ísland, sem það hefir síðan heitið

Then Floki went up on a hight mounatin and looked north over the the mountain to the fjord full of ice; from this they called the land Iceland, as it has afterwards been called.

Íslendingabók does not provide this story, but there are no conflicting narratives on the origins of the term, so we can accept it as at least what medieval Icelanders believed, if not 100% true.

It's worth noting that ongoing research suggests potential Scandinavian awareness of Iceland significantly sooner than the late 9th century, which at least asks the question of what the place was known as prior to Hrafna-Floki. Sadly, this is not really a knowable question.

GREENLAND: Our source on the naming of Greenland comes from Eiríks saga rauða. In it, Erik the Red named the place Greenland, because:

menn þat mjök mundu fýsa þangat, ef landit héti vel.

Men will settle there more, if the land is named well.

This ignores the historical reality of Greenland - while it was less harsh in the Medieval Climate Optimum, it was by no means a pleasant place to live, and the Greenlandic Norse population was never particularly large.

In both of these cases, it remains possible that these were stories invented to give a name an origin, rather than a historical recording of the actual origins. However, we don't have any alternative explanations available, so we have no choice but to accept them as is.

Why did it continue

This is probably the easiest one to answer: At some point, it's traditional! A place gets recorded under a certain name, and then it becomes hard to change (England, for instance, goes all the way back to Gildas and Bede shortly after the conversion of Germanic Britain to Christianity). Names additionally inscribe identity and narratives about themselves - being from a place is a powerful point of commonality. This is always an important thing, and appears in the very earliest writings, but especially in the thirteenth century onward, as Iceland came under Norwegian and then Danish influence, Icelanders felt the need to retain their identity over the later Middle Ages. This goal, ultimately, remained important all the way through to the Icelandic independence movement of the 19th century, and indeed all the way to the present.