The notes I am reading cover differential geometry at only a very basic level (say around Do Carmo's Differential Geometry of Curves and Surfaces), so I apologize in advance if this question would be considered incredibly misguided for anyone with a substantial knowledge of the subject.
Here is the definition of tangent space which I have:
Let $M \subset \mathbb{R}^n$ be open. Then $TM := M \times \mathbb{R}^n$ is called the tangent bundle of $M$. For $p \in M$, $T_pM := \{ p \} \times \mathbb{R}^n$ is called the tangent space of $M$ at $p$.
$\dots$
If now $f: M \subset \mathbb{R}^n \to \mathbb{R}^m$ smooth, then we consider the differential of $f$ to be a map $df: TM \to T\mathbb{R}^m,$ and $d_p f:T_p M \to T_{f(p)}\mathbb{R}^m$.
$\dots$
Let $S \subset \mathbb{R}^3$ be an embedded surface, $q \in S$ be a point of $S$, and $f: U \to \mathbb{R}^3$ be a parametrization of $S$ such that $f(p)=q$. The tangent space of $S$ at $q$ is defined to be $T_q S := (d_p f)(T_p U) \subset T_{f(p)} \mathbb{R}^3$ and the tangent bundle $TS \subset T\mathbb{R}^3$ of $S$ is $TS := \bigcup\limits_{q \in S} T_q S$.
My question:
Why the notation $T_pU$? The $U$ here does not provide any useful information about the nature of the object. All it says is that $p \in U$, which seems almost redundant.
The most important aspect of the definition is that $T_p U \simeq \mathbb{R}^n$, but this is impossible to deduce from the notation $T_pU$. I find it very difficult to understand any discussion of tangent spaces using this notation unless ample context about the ambient spaces is given nearby, which it usually isn't.
Why would anyone care more about $p \in U$ than the fact that $T_p U \simeq \mathbb{R}^n$? What am I missing?