Are there data structures that mix a tree structure with lists?
Yes.  Build a tree (with any branching factor you want).

Then, augment the data structure by maintaining one doubly linked list per level that contains all of the nodes at that level.  In particular, in each node you have a "left" and "right" field which goes to the next and previous elements in that doubly linked list (i.e., the next sibling to the right of it and the previous sibling to the left of it).

All of the standard tree operations can be augmented to update/maintain these doubly linked lists as well.  This adds only $O(1)$ extra time to each operation to update the doubly linked lists, so the asymptotic running time for each tree operation remains the same.

This construction can be applied to any tree with any structure you want (binary tree, $k$-ary tree, b-tree, AVL tree, red-black tree, etc.)

I don't know of any particular name for this specific case.  It is an instance of augmenting a data structure.  Many data structures/algorithms textbooks will have a chapter on augmenting data structures.