As Prof. Filmus said, it isn't necessary for Binary Search Trees (hereafter referred to as BST's) to necessarily have ints/Integers as the data within the nodes.
At least in Java, all we need is data that is either Comparable to itself or some superclass of itself (implementing Comparable) or can be compared with an outside tool (using a Comparator). But I digress, as that is language-specific. It should be clear however that we only need elements to be comparable/order-able. They do not only need to be integers.
You could always, theoretically, use BSTs to model decision making, as you're simply going through Decision A or Decision B at any given point. You'd have to create Decisions that were comparable to one another, but that's fuzzy on my end.
Let me therefore take an example in your post - searching through a database for a name. Of course, names are usually encoded as Strings (a sequence of characters, informally speaking). Strings are usually ordered lexicographically, that is, by alphabetical order. This means that to retrieve a given String from a BST of Strings, we'd start at some root node which contains a root String (consider a String of this database in the M or N range, as that is the middle of the English alphabet), and if the name comes after M/N or before M/N, it will go to the right node or the left node of the root node (containing the middle name), respectively.
Theoretically, you could encode the entire English lexicon into one very large BST, if you have a word that you've set as the exact middle of the English language. Every word that isn't the exact middle word would be coming after or before said word, alphabetically speaking.
I'd point you to this link, about Legislation perhaps being NP-Complete, for some discussions about decision trees. This may be an interesting, albeit not necessarily appliable analogue with BST's: Is legislation NP-complete?
Hope this helps!
EDIT: Edited to say "Prof. Filmus..." as that is more formal.