Let $A$ and $B$ be $C^*$-algebras. Is it true that $$M(A \oplus B) = M(A) \oplus M(B)$$ Since the direct sum of double centralizers of $A$ and $B$ gives a double centralizer of $A \oplus B$, we should get an injection $$M(A) \oplus M(B) \hookrightarrow M(A \oplus B)$$ but I'm not sure how to proceed from there.
Specifically, I would like to compute the multiplier algebra of $\mathcal K \oplus \mathcal K$ where $\mathcal K$ are the compact operators.
Yes, it is true. More generally, we have $$M\left(\bigoplus_{i\in I}A_i \right)= \bigoplus_{i\in I}^{\ell^\infty} M(A_i)$$ where the direct sum on the left can be interpreted as either a $c_0$ or an $\ell^\infty$-direct sum and on the right we have the $\ell^\infty$-direct sum. See the discussion here.
In particular, if $H$ is a Hilbert space, you have $$M(\mathcal{K}(H)\oplus \mathcal{K}(H)) = B(H)\oplus B(H)\subseteq B(H\oplus H).$$