On the definition of the direct sum in vector spaces We say that if $V_1 , V_2, \ldots, V_n$ are vector subspaces, the sum is direct if and only if the morphism $u$ from $V_1 \times \cdots \times V_n$ to $V_1 + \cdots + V_n$ which maps $(x_1, \ldots, x_n)$ to $x_1 + \cdots + x_n$ is an isomorphism.
Looking at the definition of a direct sum in categories in Wikipedia, it is clear that $V_1 \times \cdots \times V_n$ can be given canonical injections so that it is a categorical sum. Thus, we see that if the $u$ is an isomorphism, $V_1 + \cdots + V_n$ with $f_i(v_i)=v_i$ with i between 1 and n as canonical injections is also a sum.
But what if there is another isomorphism between $V_1\times\cdots\times V_n$ and $V_1 +\cdots+V_n$, is $u$ always an isomorphism (true if all $V_i$ are of finite dimension) ? Is $(V_1 + \cdots +V_n,(f_i)_{i=1 \ldots n})$ still a sum (again true if all $V_i$ are of finite dimension) ?

No. It is possible that $V_1$ and $V_2$ are subspaces of some vectorspace $W$ such that the subspace $V_1 + V_2$ of $W$ is isomorphic to $V_1 \times V_2$, but the canonical map $V_1 \times V_2 \to V_1 + V_2$ is not an isomorphism.
Example. Take $W$ the vectorspace of infinite sequences of real numbers (with or without finite carrier; doesn't really matter) and take $V_1$ and $V_2$ to be equal to $W$. Then $V_1 \times V_2 \cong W = V_1 + V_2$ (by interleaving the coordinates of $V_1$ and $V_2$), but the canonical map $V_1 \times V_2 \to V_1 + V_2 = W$ is not injective.
