I thought it was to hide from people that she was learning sword fighting, but whenever she refers to her teacher as dancing master people tend to understand it was her fencing teacher — for example, when she discusses it with Jaqen H'gar.
Are fencing teachers commonly referred to as dancing masters?
Although swordplay is sometimes referred to as dancing, the term was also deliberately used to deceive others as to the nature of Arya's training.
As Ned Stark first confronts Arya with Needle in hand, he muses (from the book)
"Lyanna might have carried a sword, if my lord father had allowed it"
implying that it was not conventional for women of Westeros to learn sword fighting.
When Arya first meets the bravo Syrio, he states (from the book)
"You are late, boy"
further reinforcing the notion that girls are not taught to use weapons.
Syrio then explains the water dance (from the book)
"Remember, child, this is not the iron dance of Westeros we are learning, the knight's dance, hacking and hammering, no. This is the bravo's dance, the water dance, swift and sudden."
Later, here's the exchange between Sansa and Arya (from the book)
"How was your dancing?" "I'm sore all over," Arya reported happily, proudly displaying a huge purple bruise on her leg. "You must be a terrible dancer," Sansa said doubtfully.
implying that Sansa has no clue as to the actual nature of the "dancing lessons"
When Ned starts doubting Syrio's effectiveness and suggests alternate teachers, he again reinforces the notion that these lessons are "on the sly" by suggesting (from the book)
"Or I might have a quiet word with Ser Barristan."
As for why Jaqen H'gar understands, that's self explanatory when you know that he is also a bravo so one can assume he would be familiar with water dancing as a fighting form and the reference to it as "dancing".
Her father said he was a dancing instructor to hide what he was doing and who he was (he may have hired him as a bodyguard as well as an instructor, but there’s no real evidence aside from the fact that he protected Arya).
The teacher was teaching her to "Water Dance" I believe was the phrase, which I interpret as the name of a fighting technique.
I am a sword fighting instructor and have trained several dancers over the years. They generally make good fighters quite quickly, ballet dancers are especially good. The Spartan saying was that 'a warrior who could not dance was awkward in peace and in war' so there is a fair history of this being observed :-)
Originally he was called a dancing master to conceal what was actually happening. Sansa was horrified when Arya showed he some bruises she had got 'dancing' at dinner once. The fact that it was a dance, a water dance fighting style also meant that Stark did not have to actually lie to anyone.
I think that Arya called him dancing master out of habit. Jaqen H'gar probably knew exactly what she meant as he seems extremely astute and could deduce by context (although I do not know the exact quote you are referring to so I cannot be sure).
Mostly because you don't hire sword fighting instructors to train young proper ladies of nobility. When he was hired, they disguised what they were doing by claiming to others that he was her dance instructor. He talked about the art of swordplay as a dance, etc, but it was, mostly, at first, about keeping it on the down-low.
