So when choosing a horse and a biome as soon as you press the submit button even if you go immediately back to the menu without actually using any moves or anything it still counts as that horses daily explore.

I wouldn't exactly be calling it a bug if it wasn't for the fact that it doesn't make the player wait to explore. You just no longer have the option of the previous horse.

This probably isn't a real bug however it is something that doesn't seem quite right and I know if I was the one running the game I'd want to know 🤷‍♂️

That's how it works. One you open explore with a horse, you can't unuse it.