There seem to be two competing definitions of the direct sum of vector spaces. The first one characterises it as the same as the Cartesian product for a finite number of vector spaces, and for an infinite of vector spaces it imposes the restriction that no element of the direct sum may have an infinite number of non-zero entries. Sic Wikipedia: "If the index set is finite, the direct sum is the same as the direct product".
A second definition is that the direct sum differs from the Cartesian product even for a finite number of vector spaces. The Cartesian product of $X$ and $Y$ is identified with the pairs $(x,y)$ s.t. $x \in X$ and $y \in Y$, while the direct sum is also endowed with the following linear structure:
$$ (x_1,y_1)+_{X \oplus Y}(x_2,y_2):=(x_1+_X x_2,y_1+_Y y_2) \\ \alpha \cdot_{X \oplus Y}(x,y):=(\alpha \cdot_X x,\alpha \cdot_Y y), $$ where $+_{X},\cdot_X$ are the vector sum and scalar multiplications on $X$ (similarly for $+_{Y},\cdot_Y$). Additionally, this definition carries too the restriction that no element of the direct sum may have an infinite number of non-zero entries.
My question: is the restriction that elements of the direct sum may not have an infinite number of non-zero entries motivated by the fact that the direct sum is equipped with a vector space structure? I suspect some or another vector space axiom or nice property may fail if we allow $(x_j)_{j\in I}\in \oplus_{i\in I} X_i$ with $|I| = \infty$ and $X_i$ being a vector space $\forall i \in I$, but $|\{j \in I | x_j \neq 0 \}| = \infty$.
The direct sum of a finite number of vector spaces is (can be defined to be (1)) the Cartesian product (as a set) with the vector space operations defined componentwise.
This vector space structure can be defined on infinite Cartesian products of vector spaces, and yields a vector space. All the axioms are satisfied.
The subset of that infinite Cartesian product consisting of tuples in which all but finitely many entries are $0$ is a vector subspace. Sometimes that is the one you want to work with in a particular context.
What you name these vector spaces (direct sum, direct product, ...) is a matter of convention.
(1) In category theory you define the direct sum using the existence and uniqueness of maps with certain properties.