A geometric definition of the addition law on abelian surfaces

This must be standard, I don't have a reference but the construction is easy: let $y^2=f(x)$ be a genus 2 hyperelliptic curve with $f$ squarefree of degree $5$ or $6$. As a set the Jacobian is the symmetric square of the curve, so let $(A,B)$ and $(C,D)$ be 4 points on the curve. Generically (apart from special configurations) there is a unique $y=g(x)$ with $g$ of degree 3 which passes through the 4 points (4 linear equations in 4 unknowns). Replacing in the equation of the curve gives (again generically) a sixth degree equation, 4 of the roots being the abcissas of $A$, $B$, $C$, $D$. The other two roots define your addition law, as usual after changing the sign of $y$.

Jacobians of genus-2 curves - and abelian surfaces in general, I suppose - can be realized as the variety of lines on the intersection of two quadrics in $\mathbb{P}^5$ (once you've chosen a line to act as the neutral element). This is analogous to seeing an elliptic curve as the variety of 0-dimensional spaces (i.e. points) on the intersection of two quadrics in $\mathbb{P}^3$ (which is sometimes called the "Jacobi" model of an elliptic curve). The group law has a really nice geometric expression.

This is covered at length in Chapter 17 ("A neoclassical approach") of Cassels and Flynn's Prolegomena to a middlebrow arithmetic of curves of genus 2, and in even more length in Chapter 6 of Principles of algebraic geometry by Griffiths and Harris (specifically Section 6.3, "Lines on the quadric line complex").

Edit (bonus): If you're interested in higher dimensions, then let $X$ be the intersection of two quadrics in $\mathbb{P}^{2g+1}$, and let $S$ be the variety of $(g-1)$-planes in $X$. Then $S$ is a homogeneous space under the Jacobian of a hyperelliptic curve $C$ of genus $g$. The relationship between $X$, $S$, and $C$ (and the action of $\mathrm{Jac}(C)$ on $S$) is very explicit. Chapter 4 of Miles Reid's PhD thesis (The complete intersection of two or more quadrics) has the details.