They’re mostly for hints and costumes
Super Mario Odyssey has amiibo support, and it’s mostly innocuous. Given that you can earn the costumes they unlock in-game by acquiring enough Moons and gold, and the system itself limits you to tapping three figures at a time in five minute intervals, it’s not as overboard as Nintendo could have gone.
So here’s the rub. Basically, any amiibo known to man can be used to fuel the hint system that tells you the general location of extra Moons, the main currency of Odyssey. You pop up to three of those puppies in when talking to the Uncle amiibo character that’s typically near the start of any given world, wait five minutes, then return to reap your bounty. It doesn’t matter which world you tap/scan them into, the world you’re at when you return to Uncle amiibo will get the Moon hints. You don’t need to swap figures every time when you get three more hints, you can use the exact same figures.
From there, most of the amiibo functionality boils down to costumes. Tapping an amiibo like Waluigi will get you a purple colored suit, and so on, in addition to providing hints. The pixelated anniversary amiibo are another obvious costume trigger, as are the Smash and Mario Party figures. If you want, you can also tap a Mario series figure in-game to get a temporarily invincibility boost.
That’s really it! You can ignore the amiibo concept entirely and just slowly unlock more costumes as you get more Moons, ignore the hints and find them on your own, or just tap the same three figures to the NFC touch point over and over.