Hogwarts Legacy won't allow for saved games to be played across different platforms. A Twitter post by WB Games Support confirmed this, but did mention that you can link two platforms for rewards such as trophies and achievements.
There will be a lot of options for saving your game in Hogwarts Legacy, however, such as four separate character slots. For each of those characters, you get 5 auto-save slots and 10 manual player game slots.
<social>https://twitter.com/WBGamesSupport/status/1614016096465571855</social>
Four seems a good number for characters, allowing you to have a young wizard in each of the school's houses. The only problem here is that you can't bring those characters to different platforms.
That means when the game releases on Switch and older consoles, you'll have to start afresh if you've got multiple copies. It's not the most serious complaint, but if you're looking to pick up multiple copies it's going to be something to think about.