Perhaps there is a hypothesis missing. There are counterexamples as stated. For instance, if $k$ is $\mathbb{C}[t]$ and $A$ is the quotient ring $\mathbb{C}[t]/\langle t \rangle$, then the multiplication map, $$ \mathbb{C}[t]/\langle t \rangle \otimes_{\mathbb{C}[t]} \mathbb{C}[t]/\langle t \rangle \to \mathbb{C}[t]/\langle t \rangle,$$ is an isomorphism.

Regarding your second question, there certainly are counterexamples. For instance, let $k$ equal $\mathbb{C}[x,y]/\langle y^2-x^3 \rangle$ with quotient ring $K=\mathbb{C}(t)$, where $t=y/x$. Since $x$ equals $t^2$ and $y$ equals $xt = t^3$, then the subring $A=\mathbb{C}[t]$ of $K$ is a $k$-algebra. In fact, $A$ equals $k[t]/\langle x-t^2,y-t^3 \rangle$. Thus, we have $$A\otimes_k A = k[t_1,t_2]/\langle x-t_1^2,y-t_1^3, x-t_2^2,y-t_2^3 \rangle.$$ In particular, $t_2-t_1$ is a zerodivisor in $A\otimes_k A$. Therefore the multiplication map cannot be an isomorphism.

To guarantee that the multiplication map is an isomorphism for every $k$-subalgebra $A$ of $K$, you could assume that $k$ is a Dedekind domain. Then, automatically, $A$ is a flat $k$-algebra. That is the only *general* hypothesis that works for all $A$ simultaneously **of which I am aware.**

**Edit.** I meant to add the words "of which I am aware" in my original answer (but I got distracted by something else). I added them now.

**Second edit.** In the comments below, the OP clarifies that he wants a counterexample where, not only is $A$ not isomorphic to a $k$-subalgebra of $K$, indeed $A$ is not isomorphic to a subring of $K$ (ignoring the $k$-algebra structure). Of course there are equally simple examples of this: let $k$ be $\mathbb{Z}$, and let $A$ be $\mathbb{Z}/2\mathbb{Z}$.

**Third edit.** The OP has changed his question. Let $k$ be $\mathbb{Z}$. Let $A$ be the product ring, $(\mathbb{Z}[1/2])\times (\mathbb{Z}/2\mathbb{Z})$, with its unique structure of $\mathbb{Z}$-algebra, i.e., $$A=k[x,y]/\langle 2x+y-1,xy,2y\rangle.$$ Of course the unique ring homomorphism, $$\mathbb{Z}\to (\mathbb{Z}[1/2])\times (\mathbb{Z}/2\mathbb{Z}),$$ is injective. Distributing out the tensor product, and using that $\mathbb{Z}[1/2]\otimes (\mathbb{Z}/2\mathbb{Z})$ is zero, the multiplication homomorphism for $A$ is an isomorphism. Yet $A$ contains nonzero zerodivisors, hence it is not isomorphic to a subring of $\mathbb{Q}$.