dg.differential geometry - Divisor Intersections and Chern Class Products

I'm not sure what you mean by a line bundle on a real algebraic variety and its Chern classes, but for smooth complex analytic manifolds the Chern class of a line bundle corresponding to a divisor is Poincare dual to the homological class of the divisor, as explained e.g. in Griffiths-Harris, Chapter 1, Chern classes of line bundles. So the $c_2([D_1]oplus [D_2])=c_1[D_1]c_1[D_2]$ is indeed the indeed Poincare dual to the intersection class of $D_1$ and $D_2$. If the manifold is a surface, then we get the a number.

Maybe this not what you meant, but in that case you should really add some more details to your question.