When my husband and I got married we couldn't find much information on the internet relating to surnames, but eventually established from Home Affairs that all 3 options are available, both partners keep their own surname, partners adopt a double-barrel surname, or one partner takes on the surname of the other partner. We opted for the latter, and there wasn't a lot of paperwork to sort out at all.
What still frustrates me is that South African businesses are quite clueless when it comes to same-sex marriages. When they ask for your marital status and you say you are married, the next question is immediately "what is your WIFE'S name". A few months back my husband took out some policies with FNB Insurance, and since he listed his spouse (me) as Mr, they loaded him on their system as Mrs.