If it is branded for T-Mobile they can only be used at a T-Mobile store or to make payments on your T-Mobile account or online to buy products and services accessories. If it is a prepaid MasterCard or Visa they can be used however you wish as a debit card.
Hey, diazitzel! Generally speaking, snn_555 is correct about the limitations of a T-Mobile gift card and universal use aspect of a Visa/MC/AMEX prepaid card. Assurant may have policies in place that require a card name to deter fraud and prevent false claims, but I've not had the experience of needing to ask! Were you able to get your claim filed with them using a prepaid gift card?