What does the self-intersection number calculation in homology say?

**Update, corrected:** Here's what I mean. Consider a curve that has class `xC1 + yC2`, then its square is 

                   x^2 (C1)^2 + y^2 (C2)^2 + 2xy(C1)(C2)

which is just `2xy`. Perhaps this will shed some light on the subject.