I suggest that if you take everything you read in the books as stated facts that you are suseptible to being trolled as it should be obvious by now that GRRM presented several unreliable narrators in the book

Magical swords in Dungeon and Dragons type sci-fi are markers of sorts, and the person who wields that sword, if of unknown identity, is given clue to the reader by the origination of that sword. If anything GRRM is, he is canon D&D without variance. He started writing GoT in the late 80's as an offshoot to appease his D&D fanbase - he's played D&D himself since the game was created in the late 60's, every death in the series can be explained by using D&D dice roll math and character strengths. GRRM has defined GoT and the rest of the books in truly D&D terms. This is why he doesn't want anyone else to continue the books if he were to depart-- it's because they would screw it up. Screw it up for the fans he truly loves, his D&D crowd.

Longclaw is Jon's for a reason- it's hs due to family inheritance. Blood inheritance. The show likely won't go that way because the show doesn't understand Dungeons and Dragons nor does the show have creative minds that can synthesize information based on the Dungeons and Dragons model. The show is literally shit. When Warcraft comes out if will put the tv version of Game of Thrones to shame and a new great story will emerge of the proportions of Star Wars and Indiana Jones. HBO has completely demolished what it took GRRM 25 years to create because they fail to understand the most basic principles of Dungeons and Dragons and what that mentality is to fans.

Jon is a Mormont. The sword Longclaw is a genetic marker of sorts. Swords always identify and define the wielder.