Did you uninstall and reinstall it to be sure it wasn't corrupted?

I uninstalled and reinstalled Bitwarden but it still does not show on the menu bar.

Then you want to contact bitwarden support as to why their Addon doesn't show or give you option to list on menu bar.

Firefox 109 comes with the Extensions button in the Navigation Toolbar as the new way to manage your extensions. The panel that opens if you click the Extensions button (puzzle piece icon) serves as overflow area for extensions you install. The legacy chevron (>>) overflow area still exists but is only used for built-in toolbar buttons as found in the Customize page.

In the Extensions button panel, you can click the gear icon next to an extension or use the right-click context menu to pin the extension to the toolbar (Pin to Toolbar). To (re)move the extension from the toolbar back to the Extensions button panel, you can use "Unpin from Toolbar" in the right-click context menu.

More Information:

• https://support.mozilla.org/en-US/kb/extensions-button