Hi Everyone this is Prem, I have worked as Microsoft 365 support ambassador. We see a lot of ticket for “Another account from your organization is already signed in on this computer” issue. I have shared fixes for this issue on this post.
This issue occurs when we try to activate Office 365 application or when we try to sign in into them. This issue can appear on Office 365 application such as Word, Excel, PowerPoint, Access, etc. and occurs on a Windows computer.
So in order to fix the issue Another account from your organization is already signed in on this computer lets try series of steps :
First open any Office 365 application, eg. Microsoft Word check if you have multiple account in there signed in already. if you see any account which you do not need there, you can sign them out form there. Once signed out try to sign back in and check if you get the same error:

If you still get the same error when signing back in, try to sign back in again and make sure that the option “Allow my organization to manage my device” is unchecked or click on “No, sign in to this app only” :

This prompt appears after you have entered your email address and password. If you don’t see the above prompt, go to the settings on the computer
Open Settings on the computer > then click on Accounts :

Click on Access work or school:

If you see multiple accounts in here, you can Disconnect the accounts which you don’t want on your computer if you see the account listed as Work or School account only.
Click on Disconnect > then click on Yes.

Once disconnected, Try to sign in again in the office application and check the behaviour.
If the above steps did not not fix your issue, We can try the final step. We will try to sign in using Outlook application, if you have installed Office 365 apps from the Office portal or https://Office.com you should have outlook application installed on your computer as well.
Open Outlook > Click on File tab on the top left:

Click on Office Account :

Then Click on Switch account :

Click on Add Account :

Now Sign in using your Office 365 email address and password and check the behaviour.
These were the step which we use for this issue, do let us know if that helped you to fix Another account from your organization is already signed in on this computer issue.
Date: 24/12/2021
Author: Prem
Tags: Fix Another account from your organization is already signed in on this computer | Office 365
Great job
Please, we need more of your resolution steps… They help us alot