Does copper wire need to say it's copper?
The UL publishes a guide on wire and cable markings:  

UL Marking and Application Guide 
https://www.ul.com/wp-content/uploads/2014/04/WC_MG.pdf


  Conductor Material
  
  Compact stranded copper conductors are identified by “compact” or “cmpct,” otherwise wire and
  cable with bare or coated copper conductor material is not marked with stranding identification.
  
  If the conductor material is either aluminum or copper-clad aluminum, the product, tag or carton
  markings (depending on the product category) identify the conductor material. These markings
  will appear as “AL,” “ALUMINUM,” “AL (CUCLAD),” “ALUMINUM (COPPER-CLAD),” “CU-CLAD
  AL” or “COPPER-CLAD ALUMINUM.”
  For some wire and cable, other metals may be used as conductor material. The associated
  markings for that wire and cable are explained under the heading “OTHER.” 


The "default" is indeed copper.  

However, the AWM in the marking would lead me to believe this may be something other than building wire per NEC:  


  This guide does not address wire and cable evaluated only for suitability as factory-installed
  component wiring in other Listed equipment. Those products are Recognized by UL under the
  Component-Appliance Wiring Material (AWM) and Component-Nonshielded cable categories and
  are not identified with an NEC® wire Type designation.