What firefox extension did you install? AFAIK the supported wallet that can receive the TWT airdrop from binance is only trust wallet and it's only available for mobile users not on any browser extension.
That's why it won't show there probably because TWT isn't listed on that wallet though you can see it on any blockexplorer.
The one found in the Firefox extension store, called Binanace Smart Wallet. What I find strange is that it shows up as received in the transaction history, and if you list the assets it shows up but there is no button to send this asset elsewhere.
The reason why I didn't use the mobile app is because I got an old iPhone and my iOS doesn't support it. I guess i'll have to borrow someone's phone for the time being and get it sent that way. Strange why they don't allow TWT to be used on their extensions.