Yes, you can use Bananatag on as many desktop or mobile devices as you like. Your Bananatag account is linked to your email account, not to any one specific email device.
"Getting Started" section of your Bananatag account
Need to download another copy of the Bananatag browser extension or Outlook add-in? You can do so by logging in to your Bananatag account and clicking on the "Resources" then "Getting Started" buttons. You can also click here!