What is the definition of a subspace? I have seen people give the definition that a subspace is a vector space contained within a vector space. But is this definition actually accurate? Isn't this a special case, in particular the definition of a linear/vector subspace?
For example, I have seen hyperplanes described as subspaces, but they do not contain the zero vector (affine subspaces).
Shouldn't the correct definition of a subspace be something like this:
A subset of a set from a space such that the structure of that space still holds on the subset.
In other words, my question is whether there is a general definition of a subspace.
 A: There's not enough words in the language for all the things we want to say in mathematics.
So, words get re-used.
It's very common in mathematics for the the same word to be used for two concepts in two different contexts, particularly when there is some intuitive connection between those two concepts, even if that intuition cannot be formalized.
So for example there are subspaces of vector spaces, and there subspaces of topological spaces. Those two subspace concepts are analogous; the comment of @LázaroAlbuquerque explains the analogy in some semi-formal sense. But, if there's no particular reason to formalize the analogy --- and in this case I don't think there is any reason --- then mathematicians will not bother trying to do so.
On the other hand sometimes there are reasons to try to formalize these analogies, and for that we have category theory. For instance, one might wish to argue that the concept of a monomorphim captures the general intuition of a sub-object of any mathematical object (more accurately, an embedding of one object as a sub-object of another object). If my understanding is correct this works reasonably well for subspaces in both the vector space category and the topological space category.
A: No, you are exactly right.

Shouldn't the correct definition of a subspace be something like this: A subset of a set from a space such that the structure of that space still holds on the subset.

A term for

a vector space contained within a vector space (where the structures don't necessarily match

might be something like 'vector subset'.
There's a similar thing for groups in general:
For example, you can make a group $G$ of the set $[0,2\pi)$ and some operation (I forgot the specifics, but I believe it's described in this elementary introduction of adding angles) s.t. $G$ is isomorphic to the circle group. $G$ then is not a subgroup of $\mathbb R$. But since $G$'s set $[0,2\pi)$ is a subset of $\mathbb R$ and since both $G$ and $\mathbb R$ are groups, I like to think of $G$ as a 'group subset' of $\mathbb R$.
Differential geometry:
You can even distinguish submanifold from 'manifold subset'.
Or even just topology:
Topological subspace vs topological subset.
In category theory:
There's 'sub-object', so maybe category theory also has 'object subset' ? XD
