If the standards for what is a neural network are held consistent, then cellular neural networks should probably be considered a type of neural network.
That two adjacent layers be fully connected is not considered a requirement of what people consider neural networks. Some attention networks, most convolution kernels, and several other types of networks are not exhaustively connected.
Exhaustive connectivity is a burden on computation time that is only valuable in cases when function requires it. In cases where the parameter value for the connection can be shown to be always zero, there is no need for the potential. In cases where it is rarely nonzero, the benefit of the potential may outweigh the gain of having it.
Many kinds of networks exist that are not exhaustively connected in computer science, and biological neurons are partly connected. The axons of biological neurons might grow through an adjacent layer and connect to distant one.
In Chua's paper, it was shown that cellular neural networks converge under specific conditions, just like for other neural network types. There is no claim that they learn like humans do, and they don't use the same back propagation and gradient descent that MLPs use, but they do incrementally improve, which is the only common criteria across the many forms of artificial learning networks.
Cellular neural networks do not fit into what is generally thought of when people use the term unsupervised learning, however, they are not supervised and they do exhibit the same incremental improvement that other unsupervised networks exhibit.
Part of the issue with these names is that specialized jargon forms when only one type of a thing becomes popular, assigning the name to something more specific than the words in the name imply. Cellular neural networks may not be examples of unsupervised learning, but they learn to function better without supervision.