Brains of mammals with a folded neocortex do not have identical patterns of folding in the same species.
Do two genetically identical individuals of a species have the same pattern of brain gyrification (i.e., folds/wrinkles)? Or are there other causes?
Short Answer:
Genetically identical individuals do not have identical brain structure.
Longer Answer:
The easiest way to answer your question is to look at "natural" clones: monozygotic (i.e. "identical") twins.
Brain morphology is of a lot of interest in neuroscience because differences in brain morphology are often confounding factors in human brain studies (for example those using MRI/fMRI).
I'll refer to a study by White et al., 2002. They studied volumetric and surface morphology measures in monozygotic twins. They found remarkably strong correlations in total volume (r values of .98-.99 for measures like total brain volume, cerebrum volume, cerebral gray matter/white matter, and cerebellum), and still strong correlations with different lobes of the cerebral cortex (r values from .69 to .97 for frontal/parietal/temporal/occipital gray and white matter).
However, surface measures: surface area, gyral and sulcal curvature, and surface complexity were more weakly correlated (r values of 0.49 to 0.69).
Of course, because these are monozygotic twins raised together, it's hard to know how much of the similarities are due to the shared environment as well as shared genetics. However, it is clear that brains of identical twins are not identical, and in particular their brains seem to differ more on measures of cortical surface structure rather than overall brain size. The authors write:
Many different types of nongenetic influences may contribute to the plasticity of the cortical surface characteristics, such as educational experiences, physical activity or social interactions. Furthermore, probabilistic events during the complex process of neurodevelopment [e.g. connections between neurons (Muller et al., 1997)], differences in gene expression either by chance or modulated by early-immediate genes (Abraham et al., 1993; Worley et al., 1993), or variability in cell–cell interactions (Fletcher et al., 1991) may also contribute to variability between MZ twins.
References
White, T., Andreasen, N. C., & Nopoulos, P. (2002). Brain volumes and surface morphology in monozygotic twins. Cerebral cortex, 12(5), 486-493.
You say "copying" (question has now been edited)
Welcome to Biology.SE. CC (stands for Copy Cat) was the name of a cloned cat. We never say "copying", we say "cloning".
Cloning in nature (question has now been edited)
Note that cloning is not necessarily a human process. Many animals (included animals that have a brain but excluding all mammals) clone themselves. See wiki>parthenogenesis
Rephrasing your question (question has now been edited)
We can rephrase your question as
What is the heritability of brain gyrification?
If it is unclear why this rephrasing of your question would be correct, then you'll want to have a look at this post.
Answer
There seems indeed to have genetic factors affecting gyri (White et al. 2010). Typically, the Fibroblast Growth Factor (FGF)- and Sonic Hedgehog (SHH)-signaling pathways have been seem to affect gyri (Rash et al. 2013; Wang et al. 2016).
