A growing number of states and companies are allowing people to designate their gender as “X” instead of male or female on driver’s licenses and other forms of identification.

The changes are a response to transgender Americans who don’t identify as exclusively male or female and others who feel binary gender categories don’t accurately describe them. Two years ago, Oregon became the first state to officially accommodate such residents by allowing them to select an X in the gender field of their driver’s license to convey...