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It seems to me that there is some connection between vectors and functions. Namely, I have seen that the dot product can be defined for both, Schwarzs inequality also comes in both forms (although this is just an extension of the previous as one method of its derivation is through the dot product of think). You can talk about linearity for both; I am aware of linear transformations represented by matrices that act on vectors, and then linear differential equations for functions. You can build up linear combinations of both vectors and functions...

When I tried to look this up, I found some sources suggesting that functions are subset of vectors. Although I am still finding this very confusing.

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    You can also see vectors, at least the ones in $\Bbb R^n$, as functions $\{1,2,…,n\}\to\Bbb R$. With the counting measure, you can even write the scalar product and $p$-norms for $1\le p<\infty$ as integrals. – 2017-01-21

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Not quite sure what you are asking, but this might be an answer.

There are many vector spaces. The usual ones are $\mathbb{R}^n$ considered as $\mathbb{R}$-vector spaces. In this nice vector space your geomteric intuition holds. You can define the dot product $\cdot$ (which is an inner-product on $\mathbb{R}^n$) by $(x_1, \dots , x_n)\cdot (y_1,\dots , y_n)=\sum_{i=1}^n x_iy_i$. The Cauchy-Schwarz inequality holds which is unsurprising from a geometric point of view.

Now there are more exotic vector spaces. One such example is the vector space $C([-1,1])=\left\{f:[-1,1]\rightarrow \mathbb{R}\mid f \text{ is continuous }\right\}$ of continuous real functions defined on the interval $[-1,1]$. A function $f\in C([-1,1])$ is now called a vector since it belongs to a vector space.

Given two such functions $f$ and $g$, you can also define an inner-product by $f\cdot g=\int_{-1}^1f(x)g(x)\mathrm{d}x$. Still the Cauchy-Schwarz inequality holds. That is, $|f\cdot g|\leq \|f\|\|g\|$ where $\|f\|:=\sqrt{f\cdot f}$.