In A New Hope, Darth Vader says this to Obi-Wan:
When I left you, I was but the learner; now I am the master.
However, Darth Vader was still an apprentice, and therefore a "learner" to Darth Sidious. So what did he mean by this?
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Of course, Vader is indeed apprenticed to Sidious. From Revenge of the Sith:
PALPATINE: You are fulfilling your destiny, Anakin. Become my apprentice. Learn to use the dark side of the Force.
ANAKIN: I will do whatever you ask.
He believes that, in power and knowledge, he has now outstripped Obi-Wan:
Also from Revenge of the Sith:
ANAKIN: This is the end for you, My Master. I wish it were otherwise.
ANAKIN jumps and flips onto OBI- WAN's platform. The fighting continues again until OBI-WAN jumps toward the safety of the black sandy edge of the lava river. He yells at Anakin.
OBI-WAN: It's over, Anakin. I have the high ground.
ANAKIN: You underestimate my power!
This shows Anakin's frame of mind. He believes himself to be so far beyond Obi-Wan that it does not matter what advantages Obi-Wan has; Anakin will still win.
This is what he means when he is now the master: that he has surpassed his former teacher.
I believe that many action movies have a scene when a bad guy and his good guy former mentor meet again and fight. They might be old west gunslingers, samurai, martial artists, professional assassins, etc. Before fighting they trash talk. It would be very natural for the former pupil to claim to now be superior at his fighting craft.
It would have been silly for Darth Vader to claim he was the master and Kenobi was his willing student, because they were obviously enemies. Vader was claiming that now he much surpassed Kenobi in skill and power, making them metaphorically master and pupil.