You can and you do already. Your current signature is "sent from my iPhone using Tapatalk"
Go to the main tapatalk page where it lists the different forums. At the bottom to the right is your profile button. Click that button. Go down to signature and click edit signature. Choose customized signature. That will allow you to put whatever you want in your tapatalk signature.
Mine said something similar to yours so I just changed mine. Also you might want to login on a pc and edit your full forum profile when you get the chance.
Charles - Prometheus 16.5", Cassiopeia 19" TT