Yes, that’s right - the Anvil client/server exception system assumes that all exceptions have a single string in their args. Exceptions do not have the full range of expressiveness found in normal client-server communications.
If a server call throws an exception type you haven’t registered, you’ll just get an undifferentiated AnvilWrappedError. (This is true on the Uplink, same as on the server or client. As you correctly guess, it’s the name passed to _register_exception_type() that’s used to match them up.) Slightly awkward corners like this are why the function name still starts with an underscore 