In what category is the sum of real numbers a coproduct?

None (except trivially).

It's an elementary (though maybe not obvious) lemma that if $X$ and $Y$ are objects of a category and their coproduct $X + Y$ is initial, then $X$ and $Y$ are both initial.

Suppose there is some category whose objects are the real numbers, and such that finite coproducts of objects exist and are the same as finite sums of real numbers. In particular (taking the empty sum/coproduct), the real number $0$ is an initial object. Now for any real number $x$ we have $x + (-x) = 0$, so by the lemma, $x$ is initial. So every object is initial, so all objects of the category are uniquely isomorphic, so the category is equivalent to the terminal category 1.

If you just want non-negative real numbers then this argument doesn't work, and I don't immediately see an argument to take its place. But I don't think it's too likely that an interesting such category exists.

I wonder if it would be more fruitful to ask a slightly different question. Product and coproduct aren't the only interesting binary operations on a category. You can equip a category with binary operations (as in the concept of monoidal category). Sometimes this is a better thing to do.

For example, there is on the one hand the concept of distributive category, which is something like a rig (=semiring) in that it has finite products $\times$ and finite coproducts $+$, with one distributing over the other. On the other hand, there is the concept of rig category, which is a category equipped with binary operations $\otimes$ and $\oplus$, with one distributing over the other. Distributive categories are examples of rig categories. Any rig, seen as a category with no morphisms other than identities, is a rig category. Any ordered rig can be regarded as a rig category (just as any poset can be regarded as a category): e.g. $[0, \infty]$ is one, with its usual ordering, $\otimes = \times$, and $\oplus = +$.

Isomorphism classes of finitely generated right Hilbert modules over a II_1 factor are in a bijective correspondence with the nonnegative reals. The correspondence sends every module to its dimension. Moreover, the dimension function is additive with respect to the coproduct of modules. I believe you can obtain all reals using some form of supersymmetry (Quillen construction?), but then the addition of reals will no longer correspond to the coproduct of objects.