This story could definitely be classified as a quest if read under the guidelines of Thomas Foster's book How to Read Literature Like a Professor. As a quest, the story has several distinct elements:
1. a quester-or the person on the journey
2. a place to go
3. a stated purpose
4. trials and challenges along the way
5. the real purpose for going on the journey
The quester is the hungry boy.
The place he is going--the sea
The purpose he thinks he is going may be adventure or escape from a mundane home life--the story never distinctly says why he stows away.
Of course, the trial is obvious. He is hungry and is too proud to accept charity.
Foster's book says that the real reason behind every quest is always a journey of self-discovery. The boy must learn something about himself as a result of his journey. The question is, what does he learn?
I think that when he breaks down and cries after eating the cookies and drinking the milk he has no money to pay for, and he accepts the comfort from the blonde woman and the second helping of milk and cookies, he realizes that hunger is not a weakness, and neither is accepting the kindness of strangers. After receiving the gift, which was far more than a glass of milk, he knows that he can face the sea with confidence and peace of mind in knowing that should he need help, he can accept it, and that doing so does not make him less of a man. He learns that he does not have to be alone in the world, that kindness is abundant, and that he can give and receive it without shame.
A really good discussion of the story can be found in the critical analysis link below:
I'm also adding the link of an AP literature teacher who assigns Foster's book. Her page has a link to summaries of some of the chapters, including the chapter on recognizing a quest.