Chain fishing exists as a specific shiny hunting method in Pokemon XY and ORAS. This method involves sitting in one spot, (not moving ever,) and repeatedly fishing. If you ever cast a line out and do not feel anything in, the chain is broken. You can run away from these Pokemon. This can be maneuvered around by having a Pokemon that has Sticky Hold or Suction Cups as an ability in the front of your party, (fainted or alive.) Fishing around rocks also boosts your chance of getting a bite.
Technically, chain fishing exists in SwSh, but it works exactly the same as chain encountering any other wild Pokemon in Sword and Shield. Simply encounter and defeat as many of the same species of Pokemon that you want to shiny hunt. If you don't defeat any other species of Pokemon during this "chain," (which will mess with your shiny odds if you do,) after 500 of the same Pokemon defeated, with a Shiny Charm, your odds of finding a shiny are about 1/1300. However, 3% of the time, your odds are 1/512. It is a but confusing.
I just wanted to note that the source for CyberBear's answer is incorrect, as Chain Fishing as a shiny hunting method only exists in Generation 6, (XY and ORAS.) It does not exist in Sun, Moon, Ultra Sun, Ultra Moon, Sword, or Shield.
Source: EXPERIENCE, I have caught plenty of chain fishing shinies in XY, and have watched a lot of guides for doing such a thing.
Here is a guide for chain fishing in Gen 6.
Here is a guide for Shiny Hunting in Sword and Shield.
