The Schubert classes on G/P are the classes of the Schubert varieties, which are the closures of the Schubert cells, each of which contains a unique T-fixed point. The T-fixed points on G/P are the images of T-fixed points on G/B (since T acts on the fiber, which is a projective variety, hence itself has a T-fixed point by Borel's theorem).

Up on G/B, the T-fixed points are exactly of the form N_G(T)B/B, so indexed by the Weyl group W_G = N_G(T)/T. Down on G/P, they group together by the Weyl group W_P = N_P(T)/T, so they're indexed by W_G/W_P. Which is exactly what you observed in the G/P = Grassmannian case.

(Actually you asked about compact groups, so K/L where K is compact and L is compact of the same rank, which includes some cases like S^4 = SO(5)/SO(4) that is not of the form G/P for G complex and P a parabolic. Then there's still a basis of "Schubert classes", indexed by W_K/W_L similarly.)