From what I've seen so far, it appears that you can only have one player at a time in Breath of The Wild. When you start the game again, the new game option prompts whether or not you want to overwrite existing games!
At least in other previous Legends of Zelda one could select which slot to save it under, but this seems to have no option for this…
Is it not possible to have multiple players going through the game at different levels of progress???
Best Answer
So, as was noted by @BlueBarren, both the Wii U and the Switch have the ability to have multiple profiles. Some of us have always just gone along with a singular profile with no need to have any additional profiles...
If you want to have more than one game state saved in Breath of the Wild, we can simply make a profile for each player that would like to maintain their game state and play it as that user / profile!