In the context of your quote from tractate Shabbat, each of the 5 types of dread pertain to something much smaller and weaker afflicting and even overpowering something much more powerful.
In the case of the Leviathan, (a giant sea creature which according to many is an actual kosher fish with fins and scales that will ultimately be served at the Feast of the Moshiach during the days of the Moshiach), it is reported to be affected on its gills.
In Rashi's comment on the Kilbit to Shabbat 77b, he says that it is a sheretz that enters the ears of the fish. In fish anatomy, it's organ for hearing is behind the gill plate like is illustrated in this linked image.
At the end of Rashi's comment, there is a note pointing to Rashbam to Bava Batra 77b, who says:
באוסייה - בנחיריו של דג נכנס השרץ ונראה לי דהיינו כילבית כדאמרי' בעלמא (שבת עז: ע"ש) שלשה אימות הן חלש על הגבור אימת כילבית על לויתן:
Rashbam clarifies that this Sheretz enters via the nostrils of the fish, meaning it's breathing passage, which in fish anatomy means via the gills.
The Kilbit therefore sounds like a marine parasite like Neobenedenia which is a type of Monogenea. They are hermaphroditic, reproduces rapidly infects the gills and can be fatal to its host.
This same parasite is also discussed in Chullin 67b quoting Rav Sheshet ben Rav Idi, who calls this parasite Kukeyan. There is discussion there about how the parasite enters the fish which concludes, like Rashbam states, that it enters via the gills.
והלכתא קוקיאני אסירי מ"ט מינם ניים ועיילי ליה באוסייה תולעים דרני דבשרא אסירי דכוורי שריין
The Maharsha to Chullin 67b also associates this Kukeyan with the Leviathan which he says is a dag tahor because it has scales and fins and will be served at the Feast of Moshiach as part of the meal.
So in context, perhaps the dread of the Leviathan is that since there is only one of it (its mate was destroyed), it could be made invalid via the Kukeyan and would not be permitted for use in Moshiach’s seudah.
