Does anyone know what the real purpose of the "Anonymous" user
The real purpose is exactly what Microsoft stated. This permission is for granting users outside of your organization access to those folders. This is a standard mailbox permission in Exchange Server.
Also, is it a security risk?
Only if you somehow made that folder available to ...
Maybe tell us exactly what you'd like to do. When you say these users would like to "work from anywhere" what do you mean exactly? Why can't they use their email from anywhere now?
You don't need to move their email to Office 365 for them to have OneDrive, you just need an Office 365 subscription. You don't have to move anything.
As for email, you could ...
It sounds like this "user" is intended to be a placeholder or a way of specifying the large class of people who don't have accounts on your system.
Using this identifier ("user"), you or your staff can say, effectively, that they want to share a calendar or (apparently) portion of their mailbox with someone (effectively anyone) without requiring them to ...
You can setup your external mail flow to enter your org through Exchange onprem and get routed to the Office 365 tenant only for those mailboxes you want to bring onto the cloud. You then have to setup an internal mail flow from Exchange onprem to O365 so that internal onprem senders can send to the mailboxes online.
O365 mailboxes will use Exchange online ...
For your request, hybrid deployment is the option. As Timothy said. Hybrid is big and complex job. Since there are few users, you could consider moving all to Office 365.
Exchange Deployment Assistant