I get a notification "Your leash broke" when mounting a horse leashed to me (after dismounting & thus auto-leashing it). It confuses me a bit, is in English a "leash" considered "broken" with such an action? If not, then probably, the line is left from some olden times when the only reason for a leash to end was tearing it apart. I suppose that something like "unleashed animal" or "de-leashed animal" would be a more understandable description.
In addition, for what it worth: de-leashing by dropping the rope does not show any notification, and dismounting a horse straight into a dugout does not auto-leash the former (probably intended?).