My question is as stated in the title. It seems like a very trivial matter, but I want to know if it is customary to officially treat $T_{p}U$ as a subset of $T_{p}M$. It seems like this is true, for if $f:M \rightarrow \mathbb{R}$ is smooth, then $f$ is also a smooth function on $U \subseteq M$ so that for $v \in T_{p}U$, $v(f)$ is well-defined. This would allow us to say $T_{p}U \subseteq T_{p}M$.
On the other hand, if we are being more precise, $f$ is NOT a smooth function defined on $M$, but its restriction $f|_{U}$ is. Therefore, if we are being strict about this, we can't make the above argument.
I know that $T_{p}U$ and $T_{p}M$ are isomorphic, and the two sets are often "identified" through that relation, but I am wondering if I can officially treat one as a subset of the other.
Thanks!
Strictly speaking, the sets $T_pU$ and $T_pM$ are disjoint. However, as Travis Wilse pointed out in his comments, there is a natural isomorphism $T_p\iota : T_pU \to T_pM$.
Thus, for all practical purposes it will do no harm to write $T_pU = T_pM$ simply because we have a canonical identification of these objects. In that sense we can also write $TU = TM \mid_U \subset TM$ for the tangent bundles.
In the literature you will find many examples of such an abuse of notation. Don't worry, it is not dangerous.