I would like to ask a question, but not have the same account as on for example Stackoverflow. Because I feel some shame for what I would like to ask. But the default option is using the same account. I can open incognito, but that requires another e-mail, so is there a built-in solution?
There is an option you might find useful.
This method allows you to create an account on this site and ask question anonymously.
Create an account here (which you already did).
Go to Edit Profile and Settings.
Change the username and click "Save Changes".
Now go to Hide Communities and click on Hide button on every community.
After this your account will get hidden from everyone except you and moderators.