This is an excellent question. It seems especially pertinent because this one story by O. Henry gets assigned so often to so many students. It is a "feel good" story written around Christmas time to appear in a Christmas issue of a daily newspaper. One could get the impression that O. Henry wrote it with a pencil in a saloon while half drunk and under pressure to meet a Christmas deadline.
Such stories are probably not intended to be taken too seriously, and O. Henry himself does not seem to be taking his plot or his characters too seriously while he is in the process of writing it. "The Gift of the Maji" is sentimental and romantic, like the all-too-familiar movie It's a Wonderful Life, starring Jimmy Stewart, which gets trotted out every year in December to pump some Christmas spirit into the viewers so that they will feel like buying the products advertised in the commercials. O. Henry's story is also sentimental and romantic like Charles Dickens' A Christmas Carol, and not to be taken too seriously as a true picture of real life.
We might picture Jim and Della as a man without his watch and a woman who looks truly awful without her hair, living in a cheap flat and coming to hate each other because poverty and deprivation typically do not bring out the best in people. "When poverty comes in at the door, love flies out the window."
Love in a hut, with water and a crust,
Is -- Love, forgive us! -- cinders, ashes, dust.
John Keats, "Lamia"
Readers in O. Henry's day were much less educated and had much simpler tastes in literature. Many never even went to high school. They not only liked stories with happy endings, but they insisted on stories with happy endings. Charles Dickens was a literary giant on both sides of the Atlantic, and the editors all wanted stories and humorous essays that sounded like Dickens. O. Henry sounds a great deal like Dickens. Some of O. Henry's many stories continue to be read because they present glimpses of what life was like in America in the late 1800s and early 1900s.
Writers like Sherwood Anderson, Stephen Crane, Theodore Dreiser, and Sinclair Lewis brought a refreshing change to American literature. They inspired writers like Ernest Hemingway, William Faulkner and John Steinbeck, among many who reacted against the saccharine sweetness of O. Henry, Bret Harte, and perhaps especially Charles Dickens.
Jim and Della do not feel real--but nobody especially wanted realism in O. Henry's era, perhaps because they had too much reality in their lives as it was. Della only had a dollar and eight-seven cents to spend on a Christmas present. Nowadays she would just use her Visa or MasterCard.
O. Henry is an intrusive author. He tells you what you should be feeling--and perhaps you would not be feeling much of anything if he were not moralizing.