No one can enroll you in MDM. However, if the device was assigned to another organization, then it could be linked to a business and thus assigned to an MDM. But first, let's address your Apple ID question.
You appear to be going to Settings > General > VPN & Device Management > Sign in to Work or School Account... Why? The only Apple IDs that work in this location are Managed Apple IDs created in Apple Business Manager (ABM) or Apple School Manager (ASM). If you are trying to use your personal Apple ID in this location, you should be getting the alerts you are. Personal Apple IDs cannot perform enrollment because they are not associated to a business or MDM. If you want to log into the phone with your personal Apple ID, then go to Settings > Apple ID. (you likely did this already when setting up the phone as you are prompted to Sign in with your Apple ID)
Next, you can check for MDM enrollment by looking in Settings > General > VPN & Device Management. Are there any enrollment profiles? Are there any manually installed profiles?
iPhones can be enrolled in an MDM in three ways: Automated Device Enrollment (requires ABM/ASM and an MDM with an automated enrollment profile), user-initiated enrollment (requires an MDM with a web enrollment form), and user-driven enrollment (requires ABM/ASM with Managed Apple IDs setup and an MDM).
If you are resetting the device and you are not prompted to enroll, then you are not doing automated enrollment. If you do not have ABM/ASM, then you can not create Managed Apple IDs and this cannot use the Login with Work or School. If you look in VPN & Device Management and there are no manually installed profiles, then you have no device management enabled.
Now, an outlier condition would be if your Apple ID is in a family sharing plan and you are under parental controls. However, this would require physical access to your device to set the parental control settings. I am sure this is not a condition to investigate based on you resetting the device multiple times.
"…the account being signed into already exists and cannot be used again…" is because you are already logged in with your Apple ID in the main Apple ID section (Settings > Apple ID)
"….Your applied does not support the expected services on this device. Contact your administrator to sign in…". is because you are trying to user Personal Apple IDs in the Managed Apple ID login section.
The first error is because you are using the same Apple ID is two places. The second error is because you are trying to login with an incompatible Apple ID (personal Apple ID, not Managed Apple ID.)
Hope this helps. If you are having hardware or function issues with the phone, I encourage visiting an Apple Store to review the issues with the device and open a case to try and resolve the problems. Nothing you described above is a problem. These are from incorrect setup attempts that likely do not apply to your use of the device as a personal device (no business association).
