Office 365 and Work controlled computers

They need local accounts for each user and each user needs to be replicated with the same username/password on the server-side.  That said, Active Directory would be most helpful here.  They can login to Office 365 with an online account, but as you said, Server 2019 doesn't know anything about online accounts, so they need accounts setup on the server and each laptop.

Of course, I find it kind of weird they are trying to map network drives, usually that is done in a domain setting, not with some workgroup setup using Office 365.  If they are using Office 365, why are they mapping network drives?

That is what I expected to do for non-domain, but the computers will not allow local accounts.  I can create them using Control UserPasswords2, but when I go to login... they are not accessible.  It will only allow online accounts.

Weird...I'm using a local account on Windows 10 Pro at home, are these Windows 10 Home?

There is a work-around:

If you’d prefer not to have a Microsoft account associated with your device, you can remove it. Finish going through Windows setup, then select the Start button and go to Settings > Accounts > Your info and select Sign in with a local account instead.

https://www.howtogeek.com/442609/confirmed-windows-10-setup-now-prevents-local-account-creation/

That option doesn't exist either.  The person who setup the computers originally chose "My Organization" owned and logged into the Office 365 account, instead of "I own it".

Sorry, I have no experience with this type of setup where Windows is only using an online account to login.  Hopefully someone else on E-E can chime in on this one.

Depending on what they are doing and the size of the office, using office365 for a small office does eliminate the need for a server.  Many times you can replace a full server with a network drive if that is even needed at all.

I have done what you have, "...shown them connecting with different credentials for mapping the drives...".  

One point of confusion for office365 that I run into is MS allows you to sign in as either an individual or work account. The same goes for OneDrive. You can have multiple OneDrive clients and that can get confusing. I know this is getting off topic for this thread, but you do want to make sure everybody is using the "For Work or School" version.

The client doesn't use One Drive.  Server does provide file sharing as well as applications.  The server is a Remote desktop server, and no domain controller involoved.

The fastest way to sort out your problem is to do a factory reset.

Afterward, do not connect to any network before you setup the computer and you can create local account just like what you did it for the setup for old school method.

That is worst case scenario, and I hope to avoid that.  Any other thoughts?

The client doesn't use One Drive

That is not really what I was getting at.  I was referring to one point of common confusion is logging into MS accounts by inadvertently choosing "Personal" at first log in instead of "Business/School" which is what you need.  Another potential point of confusion is the students created both a Microsoft account  for the local pc and an office365 account. Hopefully the office365 account is what is used.

I think I understand better what you are asking though. I just do not use RDP like this myself. But I believe what you are after is using Azure AD which should be included in the office365 account.
https://docs.microsoft.com/en-us/office365/enterprise/about-office-365-identity
https://docs.microsoft.com/en-us/windows/client-management/connect-to-remote-aadj-pc

It also looks like Azure AD is not supported with RDP https://answers.microsoft.com/en-us/windows/forum/all/office-365-account-for-remote-desktop-access/1fc7416a-de4e-4d81-81c4-98d80015db4c and therefore your solution you stated in your original question of creating user accounts on the server , "I have shown them connecting with different credentials" is most likely the option you will need to use.

Because you were not part of the original set up and there are different options that could have been used. it is worth a call into MS. Hop on the admin panel and choose support. You will get somebody to call you back in a very short time. They will do a screen share with you and walk you through the process. I lean on their help a lot. I don't have an easy way to test what you are doing myself or I would. The set up I  have used was what you already suggested.

They selected "Buisness/School" instead of "Personal".  I do not use the Azure AD, because the RDP is a must.  They used the Office 365 account or accounts at that point... unsure which.  The Office 365 was only supposed to provide e-mail hosting and desktop versions of Office to the users.

That is the problem.  The sign in with local accounts is not available via any normal means.  And the online account is not an option for the RDP server.  We use connect with different credentials to map network drives and that works.  Is it possible they used the admin account on all laptops, locking all other user accounts out of ability to add local accounts?

I am scheduled to be on site Thursday this week (Where the laptops exist).

What I would do on Thursday is initiate a ticket within the Office365 portal. They call back in 1 to 20 minutes typically sooner than later. I have found great help from them and they can also do a screen share to help. At times they have uncovered things that were not easy to see.

So I found how to disconnect the computer from the Azure services, and set a local admin account.  After restart, the computer works the way we expect, connect to Office 365 and the server!  One of the computers sadly has decided to no longer allow any logins, and a clean OS is in progress.

Under Accounts > Work or School > Disconnect the Office 365 account from all devices.  Thanks especially to Scott for a lot of useful information.