The topological space $\mathbb{P}^1_{\mathbb{R}}$ is a circle, so it has an abelian group structure, but this is not algebraic, not compatible with the multiplication you describe, and it can't be extended to $\mathbb{P}^1_{\mathbb{C}}$ (which is a sphere and thus has no topological group structure), nor does it restrict to $\mathbb{P}^1_{\mathbb{Q}}$.
More generally, it's impossible to do this algebraically: it's an extremely well known fact that a projective curve with an algebraic group structure (as addition would be) must be an elliptic curve.