There are many acceptable answers, and in such interview, providing swiftly and with confidence an acceptable answer is more important than giving the perfect answer.
Binary trees is definitively a vaild anwer. So congratulations !
However for databases, B-trees (the "B" stands for "balanced") would be even more recomended. The B-tree is a generalisation of the binary tree where each node has more than two children. This make this data structure more efficient for optimising disk read access. The structure also needs less rebalancing than binary trees, meaning again less disk write access.
If you are interested in performance consideration, this SO answer makes an interesting comparison between both structures.
Now, just for records, in some application areas there are more specialised structures such as for example R-trees for 3D spatial data, or hash tables if you consider looking for unique keys and are ready to sacrifice some space for more speed.
Edit: Some example of popular databases (not exhaustive !):
- sqllite uses b-trees (and has an r-tree extension)
- BerkleyDB uses B-trees and hash indexes
- MySQL uses B-trees and hashes (ans has r-trees as well)
- Postgresql uses B-trees, r-tree, hashes and several others
- SQLserver apparently uses B-trees as well