To make a long story short, the Work and School Account and Personal Account combination that Microsoft creates causes a lot of trouble.
I think Microsoft is aware of this trouble. Their own webpages tell users how to fix this problem (Rename your personal account), and they apparently are actively blocking certain users from using Work and School Account services (Azure) using a Personal Account email address. But there’s no easy solution to this problem, either.

How to fix the Work and School Account/Personal Account mix-up

So how did I fix the problems caused by the Work and School Account/Personal Account mix-up?

1. I “reset my password” for the Work and School Account. Which, by the way, has terrible security. Why is the limit only 16 characters?
2. I removed all possible Work and School Account aliases from my Microsoft Account. These were all the non-outlook.com aliases.
   

   1. “Sign in to the Your Info page of your Microsoft account.
   2. “Select Manage your sign-in email or phone number.
   3. “Under Account aliases, select Add email.
   4. “Get a new Outlook.com email address from Microsoft, and click Add alias.
   5. “Select Make primary next to the email address you just added in order to make it your primary alias.
   6. “Select Remove next to your work or school email address to remove this alias from your account.

   From BruceBNews.com

3. I set the primary alias to a certain-not-to-be a Work and School Account alias.

 

I’m pretty sure the accounts are still linked, as Microsoft never gave me any notice about removing Work and School Accounts, but I think “enabling” the Work and School Account by setting a password made it work.

For those that have the option, you could also try deleting your Work and School Account.To completely remove this account, you’ll need to do the following:

1. Take over as the administrator for your domain according to Microsoft
2. Create a new Admin user under the Admin section and sign in as the new admin
3. Delete the old user that shares your email address
4. Delete the Domain from Office 365

From GCITS.com

 

Symptoms of the Work and School Account/Personal Account mix-up

How did I know my problems were due to the Work and School Account/Personal Account mix-up?

1. iOS couldn’t connect to my Outlook.com account. Nothing worked. I tried at least three times to delete the account, removing all my contacts and calendars in the process, and then add the account back according to iOS official forums. Here are some of the “issues” I tried to fix:
   1. “The problem could lie in Exchange ActiveSync not having sufficient permissions“
   2. “Apple, Microsoft Working to Fix iOS 11 Mail App Issues With Outlook.com, Office 365 & Exchange Accounts“
   3. “Cannot Get Mail. The connection to the server failed.“
   4. “deleted my account and then installed it again and it still says that it can’t connect to the server“
   5. “Solution 1. Removing the accounts and re-entering the passwords. Solution 2. Sorting out the iOS.“
   6. “Unable to Verify Information“
2. Outlook 2016 couldn’t sign into my Outlook.com account, even with the correct account password or app passwords.
3. I remember trying to sign up for Microsoft Flow a while back. That was probably the start of the mix-up.

Note that you can also “merge” Personal and Work and School Accounts by adding access to OneDrive and OneDrive for PC:

See and manage your OneDrive files
You’re all set. Your OneDrive files will appear in File Explorer in the OneDrive folder. If you use more than one account, your personal files appear under OneDrive – Personal and your work or school files appear under OneDrive – CompanyName.

From Office.com

Or by adding the Work and School Account as an alias:

Adding your work or school email to your Outlook.com
If you want to use a new email address with your existing Outlook.com account, follow the instructions in this article to create an alias. This will give you an additional email address which uses the same inbox, contact list, and account settings as your primary email address. You can choose which email address to send mail from, and you can sign in to your Outlook.com account with any of your aliases—they all use the same password.

From Office.com