Is every $1$-dimensional vector space a field?

A $1$-dimensional vector space is not a field for the same reason that no vector space is a field. Multiplication of vectors with other vectors is not defined over vector spaces. So vector spaces are not even rings.

That said, given a $1$-D vector space $V$ over $F$ you can artificially define a multiplication function $\circ : V \times V \to V$ over $V$ thus. Since $V$ is $1$-dimensional over $F$, it is generated by a single vector $e$ of $V$. So any two vectors $x, y$ can be written uniquely as $x = \alpha \cdot e$ and $y = \beta \cdot e$ for scalars $\alpha, \beta$ (here $\cdot$ is the scalar action of $F$ on $V$).

So define $x \circ y := (\alpha \beta) \cdot e$. You can check that this is well-defined, commutative, has an identity and inverses, distributes over vector addition etc. To understand the motivation of this definition, notice that we can derive $x \circ e = e \circ x = x$ from it; so essentially what we are doing is that we are looking at the single basis element $e$ that generates $V$ as the multiplicative identity $1$ in $F$. Looked at this way, $V$ is isomorphic to $F$ itself.

However the reason we do not say a $1$-dimensional vector space $V$ is a field is because in general there is no unique way to turn some such $V$ into a field. For if $e$ generates $V$, then every (non-zero) scalar multiple of $e$ also generates $V$. Hence there are in general many different ways to define a multiplication function on $V$.

E.g. if you have the $1$-D vector space which is the line $y = x$ in $\Bbb R^2$, then you can look at it as being isomorphic to $\Bbb R$ in infinitely many ways. You can for instance get one isomorphism to $\Bbb R$ by identifying the vector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ with $1 \in \Bbb R$. Or alternatively, you can identify $\begin{bmatrix} 2 \\ 2 \end{bmatrix}$ with $1 \in \Bbb R$. Or you can identify $\begin{bmatrix} 3 \\ 3 \end{bmatrix}$ with $1 \in \Bbb R$. So on and so forth. And each such identification provides a different way to turn the line $y = x$ into a field.

If you want the interpretation to be natural, or canonical, or unique, then the answer is generally no.

Consider the most familiar case: real vector spaces. Suppose $V$ is a $1$-dimensional field over $\mathbb R$. Does that mean there's a natural way to identify $V$ with $\mathbb R$? Not really. For example, $V$ doesn't know which side is supposed to be positive and which side is negative. It doesn't know which of its elements are supposed to be integers. It doesn't know how to multiply or divide. In fact, for any $v\neq0$ in $V$, there is a vector space isomorphism $V\to\mathbb R^1$ that sends $v\mapsto1$.

On the other hand, a $1$-dimensional vector space with a distinguished basis is canonically isomorphic to its base field. That's kind of a silly statement, but, well, there it is. Choosing a basis adds a lot of information.

There is also the special case of a $1$-dimensional vector space $V$ over the field with $2$ elements, $\mathbb F_2$. There is a unique vector space isomorphism between $V$ and $\mathbb F_2$, because at least $V$ knows where $0$ is and where it isn't.