Suppose I have been given a vector space $\mathbb{R}^n$, but I don’t know if the natural basis vectors I get with it are orthonormal to start with. In other words, the axes I get could be oblique. Now, obviously, I could still define an inner product using the standard basis, but that would give me an “incorrect” inner product, in the sense that the angles and lengths that I get from this inner product would be wrong.
Therefore, when we define the standard inner product, we already have a “good basis” to start with. But where does this basis come from? Is this derived from some other definition of angles and then made to coincide with the inner product definition?
You speak as if there is a notion of lengths/angles which exists prior to the chosen basis or inner product. Mathematically, it is the other way around $-$ lengths/angles are defined by an inner product $-$ but geometrically motivating this, to my mind, involves exactly this naïve assumption.
We have an intuitive notion of lengths/angles in the world around us. According to this understanding in 3D, we can choose perpendicular axes, and then a unit vector on each axis (given choice of units). We can resolve any vector into components with respect to this basis, i.e. write out coordinates, and then we can ask how to compute lengths/angles using coordinates. Note the basis vectors correspond to standard coordinate basis vectors ${\bf e}_1,{\bf e}_2,{\bf e}_3$ (by definition $-$ that's how coordinates are meant to work).
An operation natural to consider in its own right, even if it's not clear at first how it relates to what I'm talking about, is vector projection. The vector projection of one vector onto another's ray, by basic geometry, satisfies $\|\mathrm{proj}_{\bf u}{\bf v}\|=\|{\bf v}\|\cos\theta$, and is linear in $\mathbf{v}$. Thus, scaling the vector projection by $\|{\bf u}\|$ yields a symmetric, hence bilinear operation, which we call the dot product $\mathbf{u}\cdot\mathbf{v}$, and it immediately follows that it is given by the formula $\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2+u_3v_3$.
This is an "inner product" $-$ i.e. it symmetric, bilinear, positive-definite $-$ but we got it from an intuitive notion of lengths/angles, not a formal definition. Formally, we start with an inner product $\langle-,-\rangle$, which induces a norm $\|{\bf u}\|=\sqrt{\langle\mathbf{u},\mathbf{u}\rangle}$ and metric $\mathrm{d}(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$ (i.e. length), and determines angles $\theta$ between vectors via $\langle\mathbf{u},\mathbf{v}\rangle=\|\mathbf{u}\|\|\mathbf{v}\|\cos\theta$.
Given an ordered orthonormal basis for a real inner product space $V$, rewriting vectors as coordinates amounts to an isometric isomorphism $V\cong\mathbb{R}^n$ which sends basis vectors of $V$ to standard basis vectors $\mathbf{e}_1,\cdots,\mathbf{e}_n$ and which intertwines the inner product / dot product.