Steps to reproduce:
Latest Thunderbird release in Windows 10 (91.5.0) does not show the app icon badge in a non-Admin user account. Changing from new mail vs. unread mail does not make a difference. (Tray icon for new mail does display.) I have a desktop where I primarily use a non-Admin account and a laptop where I use the Admin account. On the laptop with the Admin account it displays, as well as in the Admin account on the desktop computer but not in the limited user account.
App icon badge does not display in non-Admin user account.
It should display when checked under Preferences - App Icon Options.