Isn't the term vector space a misnomer? As a Physicist, I know:
A vector is defined to be an object with a single index which follows certain properties related to rotation in the space in which it resides.
But when we study abstract vector space, a square matrix i.e. a multi-array with 2 indices can also be a vector.
However, in the language of tensors we call it as a 2nd rank tensor and it is not a vector.
Then, isn't the term Vector space a misnomer?
 A: I would say that the term is not so much a "misnomer", as it is "overloaded". A vector can be used in certain circumstances in a very specific sense, but in other circumstances, it can be used more generally. It has the disadvantage of potentially causing some confusion, but it also has the advantage of being suggestively named.
Calling them "Vector Spaces" as opposed to, say, "Boogly Spaces", instantly takes your mind to arrows in the plane or 3D space. Imagine you get introduced to a Boogly Space, never having studied or heard of vector spaces before. You know that it needs a field to form a multiplication operation, for some reason. It's not like a group or ring really, since it's an operation between two different algebraic structures, so that's a bit weird. There's a lot of axioms to learn too. It's a bit hard to keep track of all of that in your head.
Think about the mental gap you would have, convincing yourself of some basic facts. For example, $(-1)v = -v$ for all Booglies $v$. What is it about this weird multiplication between different structures that makes that true? If you think about a vector being flipped in the opposite direction, it makes intuitive sense. For Booglies, it's just another fact you have to remember.
There's a whole bunch of other terminology to keep in mind too. Vectors $v, w$ are "parallel" whenever $v = kw$ or $w = kv$ for some $k$ in the field (sometimes required to be non-zero). This makes perfect sense when making an analogy to arrows, and is really easy to remember if you're thinking about arrows. You also intuitively see how it can be simplified if you know that one of the vectors is non-zero. But, if you're looking at Booglies instead, and not imagining arrows, this is not an easy or intuitive thing to remember. You also can get things like lines, or orthogonality with a bit more structure, and so many other terms that are all inspired by the vector analogy, each codifying something well known and geometric as an algebraic concept.
I'm not saying you couldn't get over it. You could put some work in, and train yourself to think of Booglies as generalised vectors. Arguably, not being totally convinced that $(-1)v = -v$ is an asset for trying to use axioms to prove this simple fact. However, I think, the access to an ingrained vector analogy quickly supersedes this. The intuition is so rich, helpful, and consistently ingrained in the terminology that it would be a terrific shame to put to waste.
