allocate (a job or duty)

Congress had assigned the task to the agency

his leader assigned him this mission

designate or set (something) aside for a specific purpose

managers happily assign large sums of money to travel budgets

classify (a person) as having a particular sex or gender, especially on the basis of their genitals at birth

not all people who were assigned male at birth identify as men

Susan's baby was assigned female

transfer (legal rights or liabilities)

they will ask you to assign your rights against the airline