Is a BST considered a data structure or abstract data type and why?  

It seem abstract in that it is and idea created and implemented with nodes  arranged a certain way.

But it did not show up in the text here:

https://en.wikipedia.org/wiki/Abstract_data_type

Here:

https://en.wikipedia.org/wiki/Binary_search_tree

they are referred to as data structure b.c. other ADTs can be created from them.

In this train of thought, can I assume that SLLs and DLLs are data structures as well.

Is is not so black and white, perhaps more of a spectrum of classification?
A binary search tree is a data structure.  The interface it supports (the list of operations you can perform on it and their semantics) is an abstract data type.

See https://en.wikipedia.org/wiki/Abstract_data_type for explanation of the difference between an ADT and a data structure.