Under normal circumstances, he would have been able to use the Force to telekenetically take the lightsaber. The problem was that someone else was also trying to do the exact same thing he was. He was fighting for control over the saber with another Force user, and he lost.
As you mention in your last sentence, there's a hint running throughout the movie that the lightsaber "belongs" to someone specific:
Namely, it calls out to Rey from it's chest in the cantina.
However, I don't think we've seen any indication before this that these weapons have any particular "knowledge" or "loyalty" to their owners; we've seen Force users steal them from each other and use them, and even the one in question has had multiple owners.
More that likely, it's just an indication that Kylo's control over the Force isn't as good as he thinks. The lightsaber's tentative movement is very reminiscent of how one behaves in Empire Strikes Back when Luke is trying to pull his from the snow.
Throughout the movie, Kylo shows hints that he's not very polished in his use of the Force: even his lightsaber is very rough. We also see him lose his temper a lot.
is starting to accept who they are, and genuinely believes they deserve to have the lightsaber. Even if the object itself has no loyalty to an owner, the intentions of the people involved have a real effect on their use of the Force.
This was simply a case of Kylo having a battle of wills over the lightsaber with someone who was "more deserving" of it, and losing.