Does UL approved mean fire resistant?

Not necessarily ...

It depends on the product and the safety standards involved. UL approved simply means that the testing agencies Underwriters Laboratories (UL) or Underwriters Laboratories Canada (ULC) tested the product against safety standards. Underwriters Laboratories tests all sorts of products for all sorts of safety hazards. Presumably, if fire is a safety concern for the specific product tested, the approval would mean that the product met some sort of fire resistance standards.