Q:

「しまう」 as standalone words in dialogue and description, in quotes

In the story I'm reading the author use 「しまう」 as isolated word, in quotes:

中学一年の姉は、今シェイクスピアに凝っている。最初は日本語訳を片っ端から『しまって』いたが、だんだんそれではあきたらなくなり、最近ではついにオリジナルで『しまい』始めたのだ
光紀{みつのり}は順調に日本の古曲を『しまえる』ようになると、「じゃあ、これはどうだ。『しまえる』か？」と、ある日こっそり父の貴世誌{きよし}が持ってきたのは、なんと歌の譜面であった
今日もへとへととだ。毎日誰かに会いに出かけているか、必ずうちに誰か来て熱心に話しこんんでいる。ここに来てからというもの、ほとんど両親と話をしていないような気がする。常野にいる時はこんなではなかった。毎日、二人でかわるがわる、彼が『しまって』いる物語について熱心に教えてくれた

I'm not sure about their meaning, and why they are in quotes.
As far as I know, 「しまう」 means "To finish, To stop, To do completely", so I was wondering about it showing the ability of the protagonist to play those music pieces. So, the second quote would mean something on the line of "When Mitsunori became able to play well that old Japanese music, one day his father Kiyoshi secretely brought him songs music scores and asked him "So, what about these? Are these 'I can do'?" (not sure about 「なんと」's meaning).
If I'm right, the other sentences should mean something like:

最近ではついにオリジナル『しまい』始めたのだ: "Lately he had suddenly started to be able to play original pieces";
二人でかわるがわる彼が『しまって』いる物語について熱心に教えてくれた: "The parents in turn told with enthusiasm to the boy stories in which he was able to do [what he was doing]".

Am I right? If so, why the quotes?

A:

As you know, this story is about a family who possesses a one-in-a-ten-million (and almost unhuman) kind of 'photographic' memory.  Its members are capable of thoroughly memorizing massive and endless amounts of literature. 
To describe their own special ability or action, they needed a word to use among themselves or use a common existing word for their own personal meaning.  That word happened to be 「しまう」, which already had the meaning of "to stow away".  You memorize 800 pages of information within days and stow it away in your brain just like an average person stows away a CD or something. 
(You seem to be thinking of the meanings of 「しまう」 when used as a subsidiary verb.)
For the highly special, specific and private meaning of the word, therefore, the author had no choice but to put it in quotation marks.
For most native speakers, however, 「しまう」 should be a very natural word choice for the purpose.  I never found it strange even for a moment while reading your question.  Had the author selected a big Sino-loanword instead, however, I would have felt less "comfortable".  That is the power of kana words.   

