Why in the union $\bigcup_{p \in U}T_p^*(\mathbb{R}^n)$ all of the sets $T_p^*(\mathbb{R}^n)$ are disjoint?

Unfortunately Tu has not given a precise definition of the tangent space $T_p(\mathbb R^n)$ when he introduced $T^*_p(\mathbb R^n)$ - and this is a source of confusion.

The tangent space $T_pM$ is properly introduced in Chapter 3 "The Tangent Space". Tu defines it as the set of all derivations $d : C^\infty_p(M) \to \mathbb R$. Here $C^\infty_p(M)$ is the algebra of germs of $C^\infty$ real-valued functions at $p \in M$. These algebras are pairwise disjoint for the points $p \in M$, thus also the $T_pM$ are pairwise disjoint and so are their dual spaces.

But let us come back to Tu's preliminary definition of $T_p(\mathbb R^n)$ on p. 10.

In calculus we visualize the tangent space $T_p(\mathbb R^n)$ at $p$ in $\mathbb R^n$ as the vector space of all arrows emanating from $p$. By the correspondence between arrows and column vectors, the vector space $\mathbb R^n$ can be identified with this column space. To distinguish between points and vectors, we write a point in $\mathbb R^n$ as $p = (p_1, . . . , p_n)$ and a vector in the tangent space $T_p(\mathbb R^n)$ as $$v = \left[ \begin{array}{rrr} v_1 \\ . \phantom{.} \\ . \phantom{.} \\ . \phantom{.} \\ v_n \\ \end{array}\right] \phantom{xxx} \text{or} \phantom{xxx} \langle v_1,\ldots,v_n\rangle .$$ ....
Elements of $T_p(\mathbb R^n)$ are called tangent vectors (or simply vectors) at $p$ in $\mathbb R^n$ . We sometimes drop the parentheses and write $T_p\mathbb R^n$ for $T_p(\mathbb R^n)$.

To be honest, this is extremely unclear. Does he mean $T_p(\mathbb R^n) = \mathbb R^n$, differing perhaps in notation by using tuples and column vectors? I do not think so. Tu speaks about the vector space of all arrows emanating from $p$, thus it should be interpreted as $T_p(\mathbb R^n) = \{ (p,v) \mid v \in \mathbb R^n \} = \{p\} \times \mathbb R^n$. These are again pairwise disjoint and so are their dual spaces.

Note that Tu explains on p. 11 that tangent vectors $v$ give us directional derivatives $D_v$ which prepares the abstract definition in Chapter 3.

You may like to have a look also at

How Can the Vector Space $\mathbb{R}^n$ be identified with the Column space

Directional derivatives at $P$ are all derivations at $P$

Proof of Isomorphism between Tangent Space and the Vector Space of all Derivations

Why is the tangent bundle defined using a disjoint union?