The North American website for Splatoon 3 shows that the game supports cloud saves, but it seems there’s a bit more to the story than that.
If you hop over to Splatoon 3’s eShop listing, you’ll also see a mention of cloud saves. That mention includes a caveat though, as it says this cloud save ability only pertains to offline play. While most play the Splatoon series for online play, there is indeed some lengthy single player content, and it seems that’s where the cloud save compatibility comes in.
While it’s disappointing that Splatoon 3 doesn’t have blanket cloud saves, this is a step up from Splatoon 2, which didn’t include support for the function at all. Maybe once Splatoon 4 rolls around, we’ll get full-on cloud saves!