If you have an inner product space $\left(E, \varphi\right)$, it has a natural structure as a normed vector space: $\left(E,x\mapsto \sqrt{\varphi(x,x)}\right)$ but the other way around isn't true. There are norms that do not come from inner products.
And example with $E=\Bbb R^2$
If you take $\varphi:\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)\mapsto x_1x_2+y_1y_2$ you have an inner product.
And if you let $N_2:\left(x,y\right) \mapsto \sqrt{\varphi\left(\left(x,y\right),\left(x,y\right)\right)}=\sqrt{x^2+y^2}$, you get the norm you know.
But there are other norms such as $N_\infty:(x,y)\mapsto \max(x,y)$ that can't be built from an inner product.
By the way, if your norm $N$ does come from an inner product, you can get the inner product back by letting $\psi:(x,y)\mapsto \cfrac{N(x+y)
^2-N(x-y)^2}{4}$
If your space is finite dimensional, then it is basically $\Bbb R^n$ (or $\Bbb C^n$, if it's over the complex numbers).
Suppose that $(v_1,\dots,v_n) \subset \Bbb R^n$ is your "orthonormal" basis of choice. We can define a linear transformation $M$ by
$$
M:v_i \mapsto e_i
$$
where $\{e_i\}$ is the standard basis. We can then define the inner product by
$$
\langle x, y\rangle_M = \langle Mx, My\rangle = y^T(M^TM)x
$$
We have to be a little bit more careful with infinite inner product spaces.
In particular, suppose that our space is a Hilbert space with a countable Schauder basis (so, $V = \ell^2$ up to isomorphism).
Suppose that we have a bounded sequence $(v_i) \subset V$ such that $\inf_i\|v_i\| > 0$. Then we can define a bounded linear map
$$
M:v_i \mapsto e_i
$$
So, we can define the inner product
$$
\langle x, y\rangle_M = \langle Mx, My\rangle
$$
and, because $M$ is an isomorphism of vector spaces, the metric induced by this $M$ inner product is equivalent to the usual metric on $V$.
Best Answer
It's perfectly fine to define both structures on some ring. A well-known example of such an object are the $\newcommand{\b}{\mathbb} \b{R}$-algebra of complex numbers with the conventional multiplication \begin{align} \b{C} \times \b{C} &\to \b{C} \\ (z, w) &\mapsto zw \end{align}
and the inner product
\begin{align} \b{C} \times \b{C} &\to \b{R} \\ (z, w) &\mapsto \operatorname{Re}(\bar{z} w) \end{align} Using polarization, the inner product is equivalent to the norm $|\cdot|: \b{C} \to \b{R}$. All these structures have been known to be useful.
However it should be noted that, as is usual when defining multiple different structures on an object, these structures on $\b{C}$ are so useful, because they are compatible. In this particular case compatibility takes the form of the multiplication rule for absolute values $|z w| = |z| |w|$.
Other examples I can think of: