Why does the subspace need to go through the origin?

A subspace is a vector space, then it must satisfy all axioms for a vector space, including the existence of a zero vector.

You want to be able to operate vectors in the subspace without leaving it. If any vector $v$ is in there you want $(-1)v = -v$ also to be there, and also their sum $v+(-v) = 0$.

You can indeed have two vector spaces over the same field $F$ such that the identities are different (i.e. does not go through the origin). However, these two vector spaces would simply be different since their additions and multiplications would necessarily have to be different as well. By thinking of the "origin" you are already implicitly referring to that one vector space of $\mathbb{R}^n$ with the usual $+$ and $\cdot$.
An example of this idea is already touched upon by WorldSEnder's comment above.

As for why you need some restrictions for one vector space to be a subspace of another, others have already provided very good answers.