I am trying to find the solution to $X_1$ and $X_2 \in\mathbb{R}^{n\times n}$, which satisfies the following matrix equation:
$$A_1 X_1 + X_1 A_1^\dagger + A_2 X_2 + X_2 A_2^\dagger = C_1 $$ $$B_1 X_1 + X_1 B_1^\dagger + B_2 X_2 + X_2 B_2^\dagger = C_2 $$
where $A_1, A_2, B_1, B_2, C_1, C_2 \in \mathbb{R}^{n\times n}$ are symmetric positive definite matrices.
From the first equation, notice that if we have $A_2 = 0$, we have a Lyapunov equation to which the question is addressed. More generally, when $A_2 \neq 0$, we have a simultaneous equation of "2D Lyapunov equation". Can I recast this problem into a normal Lyapunov or Sylvester-type equation?
I don't see an obvious way one could recast your problem into a Sylvester equation. However, you can still write your problem as a system of linear equation using a technique which can be used to also solve a Sylvester equation. Namely, by using vectorization and the Kronecker product. Using this your two equation can also be written as
\begin{align} (I \otimes A_1 + A_1 \otimes I)\,\text{vec}(X_1) + (I \otimes A_2 + A_2 \otimes I)\,\text{vec}(X_2) &= \text{vec}(C_1), \\ (I \otimes B_1 + B_1 \otimes I)\,\text{vec}(X_1) + (I \otimes B_2 + B_2 \otimes I)\,\text{vec}(X_2) &= \text{vec}(C_2), \\ \end{align}
with $I$ the $n \times n$ identity matrix. This can also be combined into the following system of linear equations
$$ \begin{bmatrix} I \otimes A_1 + A_1 \otimes I & I \otimes A_2 + A_2 \otimes I \\ I \otimes B_1 + B_1 \otimes I & I \otimes B_2 + B_2 \otimes I \end{bmatrix} \begin{bmatrix} \text{vec}(X_1) \\ \text{vec}(X_2) \end{bmatrix} = \begin{bmatrix} \text{vec}(C_1) \\ \text{vec}(C_2) \end{bmatrix}, $$
which can be solved using any system of linear equations solver.