Naively, most people think of vectors as "$n$-tuples" of something.
But really, a vector space is composed of 2 things:
1) stuff we can add together (these are the vectors),
2) stuff we can use to "stretch/shrink" the vectors (these are the scalars).
Formally, a vector space $V$ over a field $F$ is an abelian group:
$(V,+)$ together with an operation:
$\cdot :F \times V \to V$ with:
1) $\alpha \cdot (u + v) = \alpha \cdot u + \alpha \cdot v, \forall \alpha \in F, u,v \in V$
(the dot is usually omitted to avoid confusion with the "dot product", I just want to call attention to the fact that there is an operation here).
2) $(\alpha + \beta)\cdot u = \alpha\cdot u + \beta \cdot u, \forall \alpha,\beta \in F, u \in V$
These two conditions tell us the "scalar product" is compatible with the vector addition and the field addition.
3) $\alpha \cdot (\beta \cdot u) = (\alpha\beta)\cdot u, \forall \alpha,\beta \in F, u \in V$
4) $1_F \cdot u = u, \forall u \in V$
These two conditions tells us that the scalar product is a kind of multiplication compatible with the field multiplication.
Now in a (square) matrix ring with entries in a field, the scalar multiples of the identity matrix act just like the underlying field (a ring, by the way, is pretty much like a field but with no division...often (but not always) because it has "more zero-like things"...with matrices these "zero-like things" are called SINGULAR matrices).
In such a matrix ring, we can "keep the scalar multiplication entirely in the ring" by DEFINING the scalar multiplication to be:
$\alpha M = (\alpha I)(M)$
On the LHS, we have a "vector-looking" scalar product, on the RHS, we have a product of 2 matrices (in the ring).
There is nothing special about matrix rings in this regard...for example, we have an algebra of polynomials as well:
1) we can add polynomials
2) we can multiply polynomials by a number (field element) <--a scalar multiplication
3) we can multiply polynomials together (the "ring multiplication")
In THIS algebra, the constant polynomials play the role of the embedded field in the center (unlike matrix multiplication, this multiplication is commutative, which makes polynomials "nicer" to work with than matrices).
So, to recap:
IN an algebra (which is what the set of all 2x2 real matrices is), we have a number of different things going on:
1) We can add, subtract, mutliply and (for non-zero elements) divide field elements (the matrix entries)
2) We can add (or subtract) matrices together
3) We can multiply two matrices together
4) We can use a field element to "scale" the matrix
and all these different things work together harmoniously, to create a very satisfying structure that let's us use tools of abstract algebra, arithmetic or geometry as ways of gaining insight.
In your last sentence you seem to have it backwards: the scalar multiples of the identity act as the field, and it is the 2x2 matrices that act as the vectors. This let's us "throw away" our notion of some kind of "hybrid multiplication" (mixing scalars and vectors), and just keep the single matrix multiplication as the one we use. This streamlines having to keep track of "what came from where".