$\def\Rbb{\mathbf{R}}$ Let $F$ be a subfield of the field $K$ and let $V$ be an $n$-dimensional vector space over $F$. Then $K\otimes_FV\cong K^n$.
Considering the case when $F=K=\Rbb$ and $V=\Rbb^1$, I have $$ \Rbb\otimes_\Rbb\Rbb^1=\Rbb^1(=\Rbb)\;. $$
On the other hand, let $V$ and $W$ be two (finite-dimensional) vector spaces over the field $F$. Then $$ V\otimes_F W\cong L(V,W;F)\;, $$ where $L(V,W;F)$ denotes the set of all the bilinear maps from $V\times W$ to $F$.
If I let $V=W=\Rbb^1$, and $F=\Rbb$, then $$ \Rbb^1\otimes_\Rbb\Rbb^1 \cong L(\Rbb^1,\Rbb^1;\Rbb) \cong \Rbb^2 $$
But $\Rbb^1$ cannot be isomorphic to $\Rbb^2$. What is going wrong here?
I guess one cannot "identify" $\Rbb^1$ and $\Rbb$ when talking about tensor products since one is a "vector space" while the other is a "field".