The definition of a vector field in the standard multivariable calculus setting is the following
Given a subset $U \in \mathbb{R}^n$, a vector field is represented by a vector-valued function $f:U \to \Bbb R^n$ which is continuous if the component functions are.
It's been bugging me that the way these are presented is essentially as if the output of $f$ is attached to the input of $f$, but that doesn't really make sense in my mind.
For example let $U \subset \Bbb R^2$ and define $f:U \to \Bbb R^2$ as $f(x,y)=(1,2).$ The visualization for this map is done by attaching the vector $( 1, 2)$ to every input $(x,y)$ for $f$, but I have hard time believing that this is true. This map simply maps each point $(x,y)$ to the vector $(1,2)$ starting from the origin. Or I could simply think about this as a constant map in which everything is mapped to the point $(1,2) \in \Bbb R^2$.
What is wrong with my interpretation? Any clarification for this would be nice. I might have missed something very obvious here.
Your interpretation is totally right and better than that definition. Vector fields attach, to every point $(x,y)$, a vector originated in $(x,y)$. This is the correct geometric way to picture it, but you can see that since the "codomain" depends on the point $(x,y)$, you can't define it as a map (maps have fixed, definite codomains).
One trick to solve it is to define the space $T_{(x,y)}\mathbb{R}^2$ of vectors with origin at $(x,y)$ (this is called the tangent space at $(x,y)$) and then define $TU$ as the disjoint union of all the $T_{(x,y)}\mathbb{R}^2$ with $(x,y)$ varying in $U$. This is the space all vectors with origin in any point of $U$. That way, a vector field is a map $f:U \rightarrow TU$ such that $f(x,y)$ belongs to $T_{(x,y)}\mathbb{R}^2$ for every $(x,y)\in U$.
Since every $T_{(x,y)}\mathbb{R}^2$ is a vector space of dimension $2$, they are all isomorphic to $\mathbb{R}^2$, and that is why usually there is no distinction made between them, and a vector field is just defined as a map to $\mathbb{R}^2$. The concept of tangent space and of the tangent bundle $TU$ is introduced in differential geometry, where there's no natural isomorphism between each tangent space and $\mathbb{R}^n$.