Does Magnus Carlsen have any openings/opening variations named for him?
The last world champion I know of to have an opening variation named for him was Anatoly Karpov (e.g. Caro-Kann, Karpov Variation or Karpov Variation in the Ruy Lopez).
In the old days variations were named after players who did serious theoretical work and/or played one line consistently over a longer period.
As openings got more and more analyzed it got increasingly more difficult to find new variations comparable to those old masters. Nowadays the top players might find a novelty around move 20 or so. If you want, you could call those lines variation and credit them to the player. I am sure some websites do just that, but personally I see a big difference between say the Najdorf variation and those side lines.
So yes, probably somebody somewhere called something a Carlsen variation, but whether you consider it a variation is up to you. Also, perhaps it needs some more time until a line is called variation and maybe in 10 years people will agree to call a line Carlsen variation.
The following variation:
[FEN ""] 1.e4 c5 2.Nc3 d6 3.d4 cxd4 4.Qxd4 Nc6 5.Qd2!? Nf6 6.b3!?
Was recently introduced by Carlsen in Carlsen - Wojtaszek, Gashimov Memorial april 2018. It's currently becoming popular with other grandmasters and Chess Publishing has started referring to it as the Carlsen Variation.
If it turns out to be more than a fad for this season, I think that name will stick. Nobody played it before and the world champion introduced it to win a game in a top tournament.
Also, after Karpov, Kasparov has a variation in the Queen's Indian (1.d4 Nf6 2.c4 e6 3.Nf3 b6 4.Nc3) and Kramnik has a weird variation in the Sicilian (1.e4 c5 2.Nf3 e6 3.c4) I didn't find anything for Anand and Carlsen, though.