Let $S$ be a Riemann surface, with an atlas $(U_i, \varphi_i)_{i \in I}$. For any $P \in S$, denote $$\frac{dz_i}{dz_j}(P):= (\varphi_i \circ \varphi_j^{-1})^\prime (\varphi_j(P)).$$
We then define a meromorphic differential $\omega$ as a collection of meromorphic functions $\{f_i : U_i \rightarrow \mathbb{C}\}$ satisfying the compatibility condition $$f_j = f_i \frac{dz_i}{dz_j}$$ in the intersection $U_i \cap U_j$.
What is the meaning of this compatibility condition? What property is preserved under change of coordinates, and why give it the name differential?
Also, is this a differential form?
You may think it this way: If you want to define a function $f$ on $S$, you can just define a family of functions $f_i : U_i \to \mathbb C$ so that they agree on the intersection. That is, $f_i = f_j $ on $U_{ij} = U_i \cap U_j$ for all $i, j$.
Similarly, if you want to define a (complex) differential forms on $S$, you can define locally on each $U_i$ a one form $f_i dz^i$, so that they agree on the intersection $U_{ij}$. That is,
$$f_i dz^i = f_j dz^j \ \ \ \ \text{on }U_{ij},$$
which is the same as
$$ f_i \frac{dz^i}{dz^j} dz^j = f_j dz^j\Rightarrow f_j = f_i \frac{dz^i}{dz^j} $$
Thus you are right that the family of function $f_i : U_i \to \mathbb C$ which satisfies that condition can be glued together to form a complex differential one form on $S$. If all $f_i$'s are meromorphic (resp. holomorphic) for all $i$, then we call that a meromorphic (resp. holomorphic) one form (or differential) on $S$.