They're not extremely rare, it's just that they only spawn in one spot, which is where the waterfall connects to the ocean. If you stand on the cliff above the waterfall and fish underneath the waterfall, you'll eventually catch one. Sometimes you can even catch them without being able to see the fish shadow.
Edit: I don't know if this is always the case, but I remember catching all my salmon and king salmon when it was raining...
yeah its like the char, except iirc it swims further out into that water basin there. yes sometimes iv'e caught the fish there by sound not sight, when i couldn't see their shadow but heard them snag my line.
It's been like 3 years and all I have yet to catch is the king salmon and the regular salmon. My fossils are complete, my bugs are complete, and my diving creatures are complete... but those salmon are messing with me!
Don't know if rare. But like most said, you can catch them at the waterfall by the beach, not at the cliff. When you see one, it will appear huge like a Coleacanth. It doesn't sell for much, but it'll be worth it. If you don't see it on the waterfall by the beach, just go into a house and exit, or save and continue. You'll find it eventually. And a heads up. Between the days of the 1st through 15th and 16th through 30th will not change their rarity. So good luck!
It's been like 3 years and all I have yet to catch is the king salmon and the regular salmon. My fossils are complete, my bugs are complete, and my diving creatures are complete... but those salmon are messing with me!