When learning how to use the Sword Sokka found a master called Piandao (ironic name btw) who was a renowned master of the blade, living in the fire nation.
When asked whether he is willing to train Sokka he interruptingly proposed that Sokka must've been an elite fighter whence he came from, suggesting lots of able young men searched for him to be teached, presumably in an entitled manner.
Zuko, being an exceptional swordfighter living in the fire nation talking about nothing but his entitlement in the first 2 books fits that bill perfectly. Even among other fire nation soldiers Zuko was surprisingly skillful, so much in fact that he was almost able to free aang and fight several earthbender only equipped with his blades all by himself.
To add on that he most likely asked and/or pressured the master to remain silent about Zuko being his pupil, seeing as even General Zhaos was surprised to see Zuko possessing swords in his chamber. My guess is he could've behaved just like he did towards the prison guard of Irohs prison in book 3 ー being easily recognizable by his scar and using his position to get what he wants, but making clear that his ambitions are not to he talked about.
As to why, I can imagine he was initially trying to be better than Azula. Upon trying to beat his sister in skills she was clearly more talented than him at as a child, his mother gave him the advice to be himself, so the ambition to pursue the art of sword fighting, a skill Azula doesn't possess might have derived from that advice.
[ FanTheory ]