Like most pop culture representations of anything, significant liberties have been taken in the Marvel depiction of Thor. In the original Norse mythology, There is no hint that Thor's name or powers could be transferred to another.
Firstly, the myths do not depict Thor as being powered by his hammer the way you have described. To the contrary, he is stated to be an innately powerful divine entity. For example, according to the description given of megingjörð:
He has a second costly thing, best of all: the girdle of might; and when he clasps it about him, then the godlike strength within him is increased by half.
Gylfaginning, XXI. Translated by Arthur Gilchrist Brodeur, 1916
Likewise, the edda shows Thor to be mighty before his hammer was even made. In fact, the myths explains that Mjölnir was created partially because Thor could beat Loki into a pulp:
Loki Laufeyarson, for mischief's sake, cut off all Sif's hair. But when Thor learned of this, he seized Loki, and would have broken every bone in him, had he not sworn to get the Black Elves to make Sif hair of gold, such that it would grow like other hair.
Skáldskaparmál, XXXV. Translated by Arthur Gilchrist Brodeur, 1916
In other words, Thor's weapons and gear enhances his divine prowess - just as a human with a sword and suit of armour would be. But they are not the source of his strength.
Secondly, Thor is firmly the deity's a personal name in the texts, not a title that to be transferred. We see this from when Mjölnir changed owners in the aftermath of Ragnarök. According to the Gylfaginning, after the twilight of the gods, Thor's sons will inherit Mjölnir from their slain father.
And then the sons of Thor, Módi and Magni, shall come there, and they shall have Mjöllnir there . . . thus is it said:
In the deities' shrines | shall dwell Vídarr and Váli,
When the Fire of Surtr is slackened;
Módi and Magni | shall have Mjöllnir
At the ceasing of Thor's strife.
There is no hint here that either were to assume the name Thor - i.e., it's not treated as a title to be inherited or transferred.