I know vectors have two components, while scalars have only one, but vectors and scalars seem like they are much more simmilar than that. A lot of operations you may perform on vectors also work with scalars, such as addition, and multiplication.
For example, if I were to think of a scalar as a one dimensional vector, a lot of vector math would still work, such as the dot product. With this idea, I can think of $2D$ vectors as two dimensional scalars, $3D$ vectors as three dinensional scalars, and so on.
My question is, is this a correct view of vectors, or is my logic skewed?
Sort of. The right way to go, I think, is to conceptualize a new concept that reflects the commonalities, and then refine your understanding of "scalar" to distinguish the scalars from other objects that fit the more general concept.
You can reinforce the similarity by constructing the tensor algebra over your vector space: this gives the "universal" way to multiply vectors. The dot product, for example, can be seen as a tensor product followed by another map.
The relevant more general concept, incidentally, might simply be "vector". After all, the scalars are one-dimensional vectors!