Through the same legal process of becoming a legal parent of a child, just with someone who's already independent rather than dependent on the state. You pick up the relevant papers at the courthouse, fill out the information and go through the submission process and eventually see a family court for a judge to sign off on it.

A lot of step-parents do it or sometimes people do it so their inheritance is passed on to a person and not the state, there are some heartwarming compilations of the former on youtube.

in my case my adoptive parents got a lawyer, they filed the paperwork and i was adopted (along with my younger sisters)

once the adoption went through, i got a new birth certificate with my adoptive parents listed as my legal parents rather than my biological ones