If you want to know which e-mail you’ve linked to your account, you go to Profile > More > Account. It’s not the end of the rainbow, but it’s still the gold you’re looking for.
If you’ve signed up with your Apple ID, your e-mail found here likely ends with @privatrelay.appleid.com - it’s the adress generated by Apple at the time you decided to sign up with your Apple ID. This means, whenever you need to log in to Sleep Cycle, you’ll select Login With Apple. You don’t need to type a password then, but just use this method to log in.
If you’ve signed in with Google, the method is the same - you select Login With Google, and that should be it.