Is zero a scalar?

Yes, zero is a scalar.

So that people reading this might learn something, it turns out that the term "scalar" was first used by François Viète in his book Analytic Art (translated from In artem analyticem isagoge). He called two things scalar if they were proportionally related to each other (roughly speaking).

The first usage of "scalar" in English was apparently by Hamilton, although he used it to mean the "real" part of a quaternion, i.e. the $t$ in standard notation $t + xi + yj + zij$.

Yes, $0$ is indeed a scalar. But, to be clear, $\begin{bmatrix}0\\0\\\end{bmatrix}$ is not a scalar; it's a vector, the result of multiplying the scalar, $0$ by the vector, $\begin{bmatrix}a\\b\\\end{bmatrix}$, where $a$ and $b$ are real numbers.

As the other answers say, $0$ is indeed a scalar, and $0 {\scriptstyle \begin{bmatrix}a \\ b \end{bmatrix}} = {\scriptstyle \begin{bmatrix}0 \\ 0 \end{bmatrix}}$ is indeed a scalar multiple of ${\scriptstyle \begin{bmatrix}a \\ b \end{bmatrix}}$. More precisely, whatever field or ring of scalars your matrices are defined over, it surely has a zero element, which we conventionally denote by "$0$".

However, it may be worth noting that the symbol $0$ is also often used to denote a zero matrix, i.e. a matrix whose elements are all zero. Thus, one might e.g. write $AB = 0$, where $A$ and $B$ are matrices, to indicate that the matrix product of $A$ and $B$ contains all zeros. This does not usually cause any ambiguity, since it's clear from context what type of entity the symbol $0$ represents in each case. (Similarly, we often use $I$ to denote "the" identity matrix, without explicitly specifying what the dimensions of this particular identity matrix are.) Another justification for this notation is that, in the matrix ring of $n \times n$ square matrices, the $n \times n$ zero matrix is indeed the zero element of the ring.

Some authors might use e.g. boldface "$\mathbf 0$" or an underlined "$\underline{\underline 0}$" instead of a plain "$0$" to represent a zero matrix, especially if they follow a convention to highlight all matrix symbols in the same manner. Similarly, a zero vector might (in contexts where one makes a distinction between vectors and row/column matrices) be denoted e.g. by "$\vec 0$" instead of a plain "$0$". Many authors, however, especially in more advanced texts, will not bother with such conventions and will just expect you to figure out what type of object "$0$" represents from context.