Unfortunately same-sex partnerships are still not regulated in Serbia
I think her appointment for the position of PM is more about the image abroad than about genuine concern about LGBT+ rights. And she was chosen as the PM not after the parliamentary elections, but after the presidential elections where Aleksandar Vučić (the PM up to that point) was elected as the president of Serbia.
President Aleksandar Vučić has real power, not PM Ana Brnabić - the majority of both government supporters and opposition supporters think that.
The electorate of government parties is not exactly LGBT-friendly but they mostly tolerate Ana Brnabić.
The opposition obviously doesn't support Brnabić but it's split on LGBT issue - and that split also determines their perception of Brnabić. To part of the opposition that's not radical-right PM Brnabić is just a "puppet" of president Vučić, while the radical-right also has issues about her sexuality.