There is a purely geometric reason for why elliptic curves have group structure. A geometric shape which is also a group, such that the group operations are smooth maps, has to be homogeneous - it has to look the same from every point[1]. Not just that, if you have a vector at some point, there is a natural way to transport it to every other point on the shape. The only surface (curves over complex numbers are really 2d surfaces) which obeys this property is the torus[2].

[1] Why should the homogeneous property be true? Because in a group, multiplication by g, pushes the identity e to g. M_g(e)=g where. This is a continuous isomorphism of the shape. So the shape looks the same at g as it looks at a (a neigbhourhood of g looks the same as neighbourhood of e). So an 'X' or 'Y' shapes cant be groups, as there are points which are locally unique, but 'O' shape can be a group. Moreover, M_g can also push a fixed non-zero vector v at e to a vector v_g at g.

[2] The Euler characteristic of the torus is 0. A non-zero vector field has index 0. https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_the... See the special case of the sphere https://en.wikipedia.org/wiki/Hairy_ball_theorem