Typically, one doesn't just write down lists of axioms and then sees if there are enough interesting examples that satisfy them; they evolve over time, usually from a couple of very important/interesting examples that are then generalized.
For the vector space axioms, for example, it is pretty easy to motivate them because they crop up everywhere and are easily spotted, so it is "natural" (as natural as mathematics can be...) to write them down in an abstract fashion and say "now we are just going study just what follows from these axioms".
But for the axioms of a certain class of maps from a pair of vector spaces to (to make it simple) $\mathbb{R}$, namely the inner product, I don't find their motivation satisfying at all. From what I read in some books and Wikipedia everything boils down to saying: 1) It is a geometric fact that in - for simplicity - $\mathbb{R}^{2}$ the equation $$ \left\langle x,y\right\rangle =\left\Vert x\right\Vert \left\Vert y\right\Vert \cos\theta\quad\quad\left(1\right), $$
holdS, where $\theta$ is the angle between $x,y$ and $\left\langle \cdot,\cdot\right\rangle $ is defined as the dot product.
2) $\left\langle \cdot,\cdot\right\rangle $ has the properties of being symmetric, linear in each argument and positiv definite.
3) Conclusion: We should abstractly study symmetric, linear and positiv definite maps $V\times V\rightarrow\mathbb{R}$, where $V$ is a vector space.
For me, 1) and 2) aren't by far enough to say 3), since
$\bullet$ for other important examples of maps (and vector spaces $V$), the relation $\left(1\right)$, which motivated the abstract definition of an inner product, isn't applicable at all: It isn't intuitively clear what $\left\langle \cdot,\cdot\right\rangle $ and $\theta$ should be for these examples, so that we can verify $\left(1\right)$ for them, observe that in all these examples the LHS has the properties listed in 2), which would consolidate our belief that we truly have carved out an important class of mappings that is worthwhile studying in the abstract. Consider e.g. $V=C\left[a,b\right]$ and $$ \left(x,y\right)\mapsto\int_{a}^{b}x\left(t\right)y\left(t\right)dt. $$
$\bullet$ there are a ton of other properties the dot product $\left\langle \cdot,\cdot\right\rangle $ has. Why not study maps that satisfy some other geometric intuitive properties besides the ones in 2) ?
So what I think I'm searching for is a better motivation of the axioms of the inner product or for more example (that are qualitatively different from another) satisfying $\left(1\right)$.
Note bene: Trying to motivate the axioms of the inner product by its history didn't bring me much clarity: All I could find after some googling was that the definition of the dot product came from the definitions of the quaternions (see History of dot product and cosine), but going from there to defining inner products abstractly seems to be a bit stretched for me.
Symmetry, bilinearity, and positive definiteness are exactly the properties used to prove the Cauchy-Schwarz inequality. Well, there are a zillion proofs of the Cauchy-Schwarz inequality; I mean the one that proceeds by observing $0\le \|x-ty\|^2$ for all $t\in\mathbb R$, expanding to obtain a quadratic in $t$, and concluding that the discriminant of that quadratic is nonpositive (and then you fiddle with definiteness to get the equality case).
In other words, an inner product is just a map for which that proof is correct.
We want to obtain the Cauchy-Schwarz inequality in other spaces because it's a cornerstone of the linear-algebraic treatment of Euclidean geometry — you use it to prove the triangle inequality, to show that orthogonal projections are metric projections (which gets you everything you want to know about tangent planes to spheres), etc. (The equation (1) is part of all that: in this treatment, it's essentially the definition of angle. You need Cauchy-Schwarz to show that it's well-defined.)