The Universal Property of Tangent Spaces There are many different ways to define the tangent space $T_pM$ at some point $p\in M$, just like there are different ways to define the tensor product $V_1\otimes\cdots\otimes V_n $ or the cartesian product $V_1\times\cdots\times V_n$ of vector spaces. In the latter cases, all definitions have a universal property that guaranturees the existence of exactly one natural isomomorphism from one set to the other.
Does $T_pM$ have such a universal property, too?
By the way, I searched for this on the Internet and I found this post on Mathoverflow, but I understand neither the question, nor the answers, as they are very advanced (I am an undergraduate). I'm hoping that there is an easier answer like in the case of the tensor product. (For comparison, here is an explanation for the case of the cartesian product that I like.)
 A: $\newcommand{\dif}{\mathrm{d}}$
The following is inspired by the pages 362-3 of Tammo tom Dieck's book "Algebraic Topology". The definition of tangent spaces and the differential exploits the fact that we have already defined the differential of a differentiable function $f\colon U\subset\mathbf{R}^n\to\mathbf{R^n}$: The differential
$$Df\colon U\to L(\mathbf{R}^n,\mathbf{R^n})$$
is given by the Jacobian:
$$
Df(a)(x)=\partial_if^j(a)x^ie_j=\begin{pmatrix}\partial_if^1(a)x^i\\
\vdots\\
\partial_if^n(a)x^i
\end{pmatrix}=\begin{pmatrix}\partial_1f^1(a)&\cdots&\partial_nf^1(a)\\
\vdots&&\vdots\\
\partial_1f^n(a)&\cdots&\partial_nf^n(a)
\end{pmatrix}\cdot\begin{pmatrix}
x^1\\
\vdots\\
x^n
\end{pmatrix}$$

The universal property
Let $M$ be a differentiable, $n$-dimensional manifold - a manifold with an atlas such that all transition charts are differentiable. Consider the following subset of $A$ for some $p\in M$:
\begin{equation*}
    A_p:=\left\{x\in A:p\in\text{Dom}(x)\right\}
\end{equation*}
(According to the definition of an atlas, $A_p\neq\emptyset$ for all $p\in M$.)
A tangent space at $p\in M$ is a pair $(V,\phi)$, consisting of a $n$-dimensional real vector space $V=:T_pM$ and a map
\begin{align*}
\phi\colon A_p&\to L\left(T_pM,\mathbf R^n\right)\\
x&\mapsto\phi(x)=:\dif x_p
\end{align*}
with the following properties:

*

*$\dif x_p$ is an isomorphism for all $x\in A_p$. Thus,
\begin{equation*}
\left(\frac{\partial}{\partial x^1}|_p,\ldots,\frac{\partial}{\partial x^n}|_p\right):=(\mathrm{d}x_p^{-1}(e_1),\ldots,\mathrm{d}x_p^{-1}(e_n))
\end{equation*}
is a basis of $T_pM$ - the basis induced by $x$.

*We have
\begin{equation}\tag{1}
D(y\circ x^{-1})_{x(p)}=\dif y_p\circ\dif x_p{}^{-1}
\end{equation}
for all $x,y\in A_p$.

Lemma. The tangent space is defined up to a natural isomorphism:
Let $(V,i)$ and $(V',i')$ be tangent spaces at $p$. An isomorphism $f\in L(V',V)$ is called natural, if there is an $x\in A_p$ such that $f=\mathrm{d}x_p^{-1}\circ\mathrm{d}x_p'$. You can prove that there is exactly one natural isomorphism using the chain rule.

Definition - The Differential
Let $(M,A)$ and $(M',A')$ be two differentiable manifolds.

*

*A function $f\colon M\to M'$ is called differentiable at $p\in M$ if $y\circ f\circ x^{-1}$ is differentiable at $x(p)$ for some pair $(x,y)\in A_p\times A'{}_{f(p)}$. In this case, $y\circ f\circ x^{-1}$ is differentiable at $x(p)$ for all $(x,y)\in A_p\times A'{}_{f(p)}$.

*The differential of a differentiable function $f\colon M\to M'$ is defined as
\begin{equation*}
    \mathrm{d} f_p:=\mathrm{d}y_{f(p)}^{-1}\circ D(y\circ f\circ x^{-1})_{x(p)}\circ\mathrm{d}x_p\in L(T_pM,T_{f(p)}M')
\end{equation*}
You can prove that $\mathrm{d} f_p$ doesn't depend on the choice of $(x,y)\in A_p\times A'_{f(p)}$ by using the chain rule.

Let me conclude with a useful equation:
\begin{equation*}
    \mathrm{d}f_p\frac{\partial}{\partial x^i}|_p=\underbrace{(\mathrm{d}y_{f(p)}^{-1}\circ\underbrace{D(y\circ f\circ x^{-1})_{x(p)}\circ\underbrace{\mathrm{d}x_p)\frac{\partial}{\partial x^i}|_p}_{=e_i}}_{=\partial_i(y^j\circ f\circ x^{-1})_{x(p)}e_j}}_{=\partial_i(y^j\circ f\circ x^{-1})_{x(p)}\mathrm{d}y_{f(p)}^{-1}e_j}=\partial_i(y^j\circ f\circ x^{-1})_{x(p)}\frac{\partial}{\partial y^j}|_{f(p)}
\end{equation*}
