In many disciplines, including biology, image and signal processing, chemistry, sociology, medical imaging, and physics, self-similar networks describe and explain complicated systems with hierarchical or recursive structures. Self-similar network theory is one of the important areas in mathematics, which can be used to model real-world issues. Due to its universal applications, researchers have shown interest in self-similar networks. In this case, topological indices are used as numerical quantities that transform complex self-similar network structures into numerical values. We can discuss the intricate architecture of diamond fractal networks (DFNs) and square fractal networks (SFNs) by using the generalized fractal dimensions (GFD), which is newly defined by using different types of neighborhood degree-based topological indices. In this context, the neighborhood degree-based topological indices, namely the third neighborhood degree-based index developed by De ( $N{D}_e$), the neighborhood version of the hyper-Zegreb index, the neighborhood forgotten topological index, the Sanskruti index, and the neighborhood inverse sum index are derived and computed for the representative networks. Moreover, the multifractal dimension measures are calculated for all indices from the general form of the obtained neighborhood degree-based topological indices. In addition, the comparison graphs of all indices and the generalized fractal dimensions are shown and geometrically discussed for the aggregate structure of the aforementioned networks with respect to all indices for each iteration $(k\ge 3)$. Multifractal GFD spectral curves are also compared graphically with all indices at each iteration for the networks considered, and we analyze the complexity level of the networks at each iteration.