Why keep the point $p$ in $(p,v)$ if one later identifies the pair with $v$?

The basepoint $p$ remains important because it determines the linear map $Df(p)$ used in the output pair. The derivative depends on the point at which it is evaluated.

Conditions: Applies to the basepoint derivative $df_p(v) = (q, Df(p)v)$.

A vector field over $\mathbb{R}^3$ assigns to each point $p$ a vector $v$ belonging to the tangent space $T_p\mathbb{R}^3$ at that point.

Conditions: The ambient space in this example is $\mathbb{R}^3$.

The tangent space at a point $p$ is defined as the set of ordered pairs $(p,v)$ where the second entry $v$ ranges over all of $\mathbb{R}^n$. The first slot records the basepoint and the second slot records the vector part.

Conditions: $p \in \mathbb{R}^n$; $v \in \mathbb{R}^n$

The basepoint derivative $df_p$ is a map from the tangent space at $p$ to the tangent space at $q=f(p)$. It sends a tangent vector $v$ to the pair $(q, Df(p)v)$, packaging the ordinary derivative with the target basepoint.

Conditions: $U$ is an open subset of $\mathbb{R}^n$.; $f : U \to \mathbb{R}^m$ and $f \in C^1$.; $q = f(p)$.

The cotangent space at $p$ is the dual space of the tangent space at $p$. It is written as $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$ because it consists of all linear maps from the tangent space $T_p\mathbb{R}^n$ to the field $\mathbb{R}$.

Conditions: $p$ is a basepoint.