In James Baldwin’s "Sonny’s Blues," what is the significance of the statement “Now these are Sonny’s Blues”? How has Sonny made this music his own?

"Sonny's Blues" refers to the the musical rendition of Sonny's personal experiences of pain and suffering.
James Baldwin's short story is itself much like a blues song in its thematic repetition of emotion and mood. One of the key scenes in this story is that of the singers in the street revival meeting to whom Sonny listens. When he returns to his brother's home across the street, Sonny remarks that the woman whom he heard singing must have deeply suffered to sing as she did. He adds,

"It's repulsive to think that you have to suffer that much."

As he confides in his brother, Sonny explains that many people use heroin "to make something real for them." He adds that there is no way not to suffer, "but you try all kinds of ways to keep from drowning in it." After he reveals more about himself to his brother, Sonny extends an invitation to the nightclub where he is going to play piano. So, the brother joins Sonny that night and sits in a dark corner as "the room began to change and tighten." As he listens to Sonny and the other musicians, the narrator realizes that the one who creates the music "is hearing something else" from those who are in the audience.

What is evoked in him, then, is of another order, more terrible because it has no words, and triumphant, too, for that same reason. And his triumph, when he triumphs, is ours.

As the musicians perform, Creole leads the way "wailing on the fiddle," but listening especially to Sonny because 

[He] wanted Sonny to leave the shoreline and strike out for the deep water.

When a musician plays a blues song, he can improvise. The piano is an excellent instrument for such improvisation. This improvisation makes a song belong to the musician who plays it since his private expression goes into that song. Creole wants Sonny to make the song they are playing his own. Finally, Sonny takes the number they have been playing and "gets in the water" by expressing his pain and suffering in his improvisation. As he listens, Sonny's brother remarks upon "with what burning he had made it his." Now "these are Sonny's blues"; that is, the song expresses Sonny's individual sorrows, pain, and suffering.