I know that
The sum [of vector spaces] $U_1+\cdots+U_m$ is direct if every $v\in U_1+\cdots+U_m$ has a unique expression as $v=u_1+\cdots+u_m$ for $u_1\in U_1,\ldots,u_m\in U_m$.
I also recognise that
A basis $e_1,\ldots,e_n$ is defined if every $v\in V$ has a unique expression as $v=a^1 e_1+\cdots+a^n e_n$ for $e_1,\ldots,e_n\in V$.
These parallels are blatantly obvious. Obviously understanding the basis for a vector spaces is fundamentally important. However direct sums are given an equal amount of significance it seems.
Why are these sums of vector spaces specifically defined as direct? What makes them so special?
Not really, the basis definition is a special case. If $\{e_1,\ldots,e_n\}$ is a $K$-basis of a vector space $V$, then the sum $V=Ke_1+\ldots+Ke_n$ is direct, where $Ke_i =\{ke_i\mid k\in K\}$ is a 1-dimensional subspace of $V$.
Again: Those two definitions are different. The definition of basis leads directly to the above direct sum of 1-dim. vector spaces. But not each direct sum of vector spaces consists of 1-dim. spaces.