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I know that many types graphs have unique names, ie. a directed graph where each node has exactly one outbound edge is known as a functional graph. Do the graphs with exactly 2 inbound/outbound edges have a name?

Bonus: If the 2 inbound/outbound graph is named, is the nth inbound/outbound graph named?

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This is known as a $d$-regular graph (at least if the graph is undirected). If your graph is a sub-graph of a bigger graph, then it is a $d$-factor. Finally, for $d=2$ it's also called a cycle cover, since it's a collection of cycles (if it's finite). For $d=1$ it's called a matching. For $d=3$ it is known as a cubic graph.

For directed graphs it's usually not clear what is supposed to be regular, the in-degree, the out-degree or both.

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