Coming more from a computer science background, I never really studied tensor products, covariant/contravariant tensors etc. So until now, I was seeing the "tensor product" operation mostly as (what appears to be) a Kronecker product between the matrix representation (in some fixed basis) of my vector/linear operator, i.e. if I have two vectors/matrices
$$A = \begin{pmatrix}a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots\end{pmatrix}$$ $$B = \begin{pmatrix}b_{11} & b_{12} & \cdots \\ b_{21} & b_{22} & \cdots \\ \vdots\end{pmatrix}$$
Then: $$A \otimes B = \begin{pmatrix}a_{11}B & a_{12}B & \cdots \\ a_{21}B & a_{22}B & \cdots \\ \vdots\end{pmatrix} $$ i.e. $$A \otimes B = \begin{pmatrix} a_{11}b_{11} & a_{11} b_{12} & \cdots & a_{12}b_{11} & a_{12}b_{12} & \cdots \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{12}b_{21} & a_{12}b_{22} & \cdots \\ \vdots & \vdots & & \vdots & \vdots \\ a_{21}b_{11} & a_{21} b_{12} & \cdots & a_{22}b_{11} & a_{22}b_{12} & \cdots \\ a_{21} b_{21} & a_{21} b_{22} & \cdots & a_{22}b_{21} & a_{22}b_{22} & \cdots \\ \vdots & \vdots & & \vdots & \vdots \end{pmatrix} $$
In particular, if we consider $|0\rangle = \begin{pmatrix}1\\0\end{pmatrix}$ and $|1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$, then $|0\rangle \otimes |1\rangle = \begin{pmatrix}0\\0\\0\\1\end{pmatrix}$, i.e. $|0\rangle \otimes |1\rangle$ is a vector.
Now, if I look at the tensor product webpage of wikipedia, they seem to define $v \otimes w \colon= v w^T$, i.e. $v \otimes w$ is a matrix (ok, the matrix is just a reshape of the vector obtained by the Kronecher product so both of them are isomorphic, but in term of "type" isn't it a bit strange to define it like that?). But on the other hand, when $v$ and $w$ are matrices, we are back to the Kronecher product.
So here is my question: why do they define the tensor product like that for vector? Is there different "kinds" of tensors? How are they linked with tensors used in physics?