This seems strange to me. My iCloud account is my Gmail address (even though I also have an iCloud one on the account) and when I use Sign in with Apple I get an option to redirect all emails to Gmail.
But the system is indeed weird, I signed up for an account in an app with bike routes and wanted later to check it in the browser and had no idea what or how to find out what my account is or how should I sign in (could be also the app/website didn't implement this properly).
Things may differ from person to person depending on when they created their Apple account(s).
At first, before they had cloud services, you created an Apple account to purchase things from the iTunes store. You could use any email address.
Then they created MobileMe (which was rebranded to .Mac), and that came with an email address @mac.com. (I believe there were a couple of other domains you could choose instead, but don't remember what they were).
That was eventually discontinued, replaced with iCloud, and .Mac accounts were migrated.
Somewhere in all there Apple loosened requirements so that you could use an outside email address as your cloud ID, and made it so a cloud account could could also work as an iTunes account.
For those who created their accounts after that point, it's all sane. Create your Apple account using your outside email address if you want, or using an Apple provided address if you prefer, and then that one account can be used for all your Apple stuff. Buying music, buying or renting video, buying apps, and the cloud stuff.
For those of us who created our accounts before all that, we ended up with an account using our outside address which has our music, video, and app purchases on it, and an account using our @mac.com address that has calendars, photos, and the like.
When they changed it so all Apple accounts could be used for everything, it got even more annoying for us. Whenever we'd see some dialog asking us to sign in to our Apple account, we'd have to guess if it wanted our music/video/app account or our cloud account. If we guessed wrong, we could end up accidentally purchasing apps or media on the cloud account.
Apple does not provide any way to transfer purchases between your accounts, so if you end up with media or app purchases on both accounts there is no way to consolidate other than purchasing duplicates.
If you are willing to do that, or if you have avoided duplicate purchases, you can kind of manually consolidate accounts. You can export calendars, contacts, and the like from your original cloud account, and import them into your original iTunes account. Same for photos, online disk space, and anything else you have on the original cloud account.
Once you've got it all in the original iTunes account, delete everything from the original cloud account, and then just make sure to never again sign into that account. Any time you see the Apple account login dialog, give the original iTunes account.
Even if you have alternate email addresses associated with your iCloud account, you are free to update it to have your external email address be the default for your iCloud/Apple ID.
Although I will note that it says you can't change your Apple ID email address if you choose an @me.com, @mac.com, or @icloud.com email address for your Apple ID, which is the first time I've seen that warning.
But the system is indeed weird, I signed up for an account in an app with bike routes and wanted later to check it in the browser and had no idea what or how to find out what my account is or how should I sign in (could be also the app/website didn't implement this properly).