Tangent bundle: $TM := \bigcup_{p \in M} T_pM$, where $T_pM = \{p\} \times \mathbb{R}^n$. So, $M$ is an $n$ dimensional manifold.
Now, letting $V = \mathbb{R}^n$. A trivial vector bundle is $E := M \times V$. I know that this is trivial because we can identify $E_p$ with $V$ via $(p,v) \mapsto v$.
However, why can't we do the same and identify $T_pM$ with $V = \mathbb{R}^n$ via $(p,v) \mapsto v$?
Locally, the tangent bundle does indeed look like a trivial vector bundle. However, this doesn't have to be the case globally. A standard example is $S^2$. By the hairy ball theorem, every continuous vector field on $S^2$ must vanish somewhere. On the other hand, a manifold is parallelizable if and only if it has a smooth frame.