Yeah, in countries where gay marriage isn't legal. Some couples choose to go through adult adoption (where you legally adopt someone who isn't a minor)
This way they now can legally share a last name, if something happens and your partner goes to the hospital you are now legally family and are allowed in (bf/gf is often not good enough to make any medical decisions), you can even put them on your insurance! Basically, it's the only way to legally be family and have some of the legal things families can do even when your relationship isn't legally recognized/protected.
Fun Fact: there is (or was, I'm not super up to date about each country) some counties where they recognize gay marriage but don't let you do it. So couples go get married in a different county and then return(or don't) back home
Adult adoption it’s commonly use by same sex partners in countries that still not allow them to marry, like Korea, China, Japan and many others, this is not only a way to become family in an emotional way but also give them legal rights, like be able to make legal and medical decisions, inheritance rights and many others.

is this procedure done often in Korea? if you can't marry legally you just adopt your partner to get them to have your surname? did i understand wrong?