Yes. They're the typical data structure backing most relation database indexes. Knowing when they will perform well (and more often, when they won't) follows directly from knowing their structure. (The example of trying to do a range search on two dimensions in the post above is an example that doesn't perform as well as — again, I find — people naively expect it to.)
Have I ever implemented one? Not yet. Do I ask for a BTree implementation or exact, low-level understanding on an interview? No.
Yes! Our database blew up in our face because we misused and had wrong performance assumptions some indexes. Colleague had knowledge of btrees (the underlying data structure of database, iirc) and database internals. Optimised the whole thing. Queries are now 500x faster
Have I ever implemented one? Not yet. Do I ask for a BTree implementation or exact, low-level understanding on an interview? No.