Some monsters can inflict Stability loss, but it's not universal.
Monsters that are particularly shocking, or that exude some kind of psychic pressure, can definitely force the investigators to test against Stability loss.
However, the main use for Stability as a stat is that you can call for the investigators to test it to keep a cool head in terrifying situations. Even if a monster can't lower the investigators' Stability, you can still call for Stability tests when it's around. Investigators that fail might run, or call attention to themselves, or accidentally safety their gun/release the magazine, or otherwise make a bad judgement call due to nerves.
You can definitely do more traditional rules and have all supernatural monsters and situations cause the players to test or lose Stability, but it's optional.