Desi: No, it's not a Harmonica, it is in fact A 'Jew's harp' or a very similar instrument played in the mouth

Depends on the singer, JJ (best known) or KK (original) -

JJ: "I was playin' soft while Bobby sang the blues, yeah."

Jew's harp might fit the lyric, the blues... not so much. imo JJ is more honkytonk than country blues. Great band though.

KK: "Was blowin' sad while Bobby sang the blues."

Blowing sad blues on a Jew's harp? How to? Blues need limber reeds that one does not have to chuff 200kPa to bend a note. That's a Hohner Marine or Blues band for country folk on a budget.