To remove access to the recording so that the owner is the only one that has access, follow the steps below.
Open Microsoft Stream by navigating to m365.cloud.microsoft/launch/stream?auth=2&origindomain=microsoft365
Open the video that you want to change access to
On the right side of the screen, you will see a button that says 'Share'

Click on the bottom option that says 'Manage Access'

Click on 'Stop Sharing' at the top of the window that pops up

Close out of the Manage access window when done

Once this is done, only the owner will have access to the recording.