Biproducts in pointed, non-linear varieties

Determine whether there exists a non-linear variety with a non-zero object X for which the canonical comparison map Y_{X,X}: X + X -> X × X is an isomorphism.

Background

In additive (linear) categories, finite sums and products coincide (biproducts). The question asks whether a non-linear pointed variety can nevertheless have a nontrivial object whose binary sum and product coincide.