In a linear algebra book, the author defines, given a direct sum decomposition $V = U \oplus W$, the projection of $V$ onto $U$ with respect to this decomposition as $P_{U,W} \in \mathcal{L}(V)$ such that, given the unique decomposition $v = u+w$ where $u\in U, w \in W$, we have $P_{U,W}v=u$. Just to play around i decided to generalize it to arbitrary (potentially infinite) direct sums:
Suppose $$V = \bigoplus_{\alpha \in L}U_\alpha$$
(I defined this as each element of $V$ having uniquely as a sum $\sum u_\alpha$ where each $u_\alpha \in U_\alpha$ and only finitely many elements of this sum are non-zero. And defined infinite sums that have only finitely many non-zero elements as the corresponding finite sum of it's non-zero elements.)
Then we could define the projection onto one of those $U_\alpha$'s in the obvious way. However after thinking about this, i noticed that, at least if the sum is finite, this construction adds nothing new: We could have an equivalent decomposition that uses only 2 terms, for example if $V = U_1 \oplus$...$\oplus U_n$ then also $V=U_i \oplus \bigoplus_{k\neq i} U_k$. Can we do the same this if the direct sum is infinite?
To get a concrete picture in mind, think of the space $X$, the direct sum of countably many copies of $\mathbb R$. This is simply (isomorphic to) the space of all eventually null sequences of real numbers. It is clear that indeed any such eventually null sequence is uniquely the finite sum of its projections to the non-null coordinates.
Now, if you single out any particular coordinate position, say the first one, and eliminate it, then you still have the space of all eventually null sequences. In other words, you get $X$ again. A moment's thought should thus show that $X\cong \mathbb R \oplus X$.
In the most general case the same phenomenon is true. The proof can be given directly in terms of the condition you gave in terms of unique decomposition (it's a straightforward exercise). Much more generally though it is a categorical principle. The direct sums you are looking at are the categorical coproducts in the category of vector spaces. First, note that similarity of the property you are asking about to associativity. You are basically asking if the brackets can be moved around in the usual way. Your question is about what happens when there are infinitely many summands. The universal property of coproducts guarantees that such moving around of brackets results in isomorphic answers. No axiom of choice is involved.