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The morphology of solid surfaces encodes fundamental information about the physical mechanisms that govern their formation. Here, we reinterpret scanning electron microscopy (SEM) micrographs of oxide thin films as two-dimensional self-affine morphology fields (not height-metrology) and analyze them using a multiscale statistical-physics framework that integrates spectral, multifractal, geometric, and topological descriptors. Fourier-based power spectral density (PSD) provides the spectral slope $\beta$ and apparent Hurst exponent $H$, while multifractal scaling yields the information dimensions $D_q$, the singularity spectrum $f\left(\alpha \right)$, and its width $\mathsf{\Delta }\alpha$, which quantify scale hierarchy and intermittency. Lacunarity captures intermediate-scale heterogeneity, and Minkowski functionals—especially the Euler characteristic $\chi \left(\theta \right)$—probe connectivity and identify the onset of a percolation-like network structure. Two representative surfaces with contrasting morphologies are used as model systems: one exhibiting an anisotropic, porous, strongly multifractal structure with fragmented domains; the other showing a compact, nearly isotropic, and nearly monofractal organization. The porous surface/topography displays steep PSD decay, broad multifractal spectra, and positive $\chi$, consistent with a sub-percolated, diffusion-limited, Edwards–Wilkinson-like (EW-like) growth regime. Conversely, the compact surface/topography exhibits gentler spectral slopes, narrower $f\left(\alpha \right)$, enhanced lacunarity at intermediate scales, and a $\chi \left(\theta \right)$ zero-crossing indicative of a connectivity transition where a surface becomes a percolating network, consistent with a Kardar–Parisi–Zhang-like (KPZ-like) correlated growth regime. These results demonstrate that individual SEM micrographs encode quantitative fingerprints of nonequilibrium universality classes and topology-driven transitions from fragmented surfaces to connected networks, showing that SEM intensity maps can serve as a quantitative probe for testing theories of rough surfaces and kinetic growth in experimental thin-film systems. Full article