The monarchy violates sexual discrimination laws because it is based on the male line, and first born sons. Because it is based on a male, a female would never be the heir, and even if she were to be in power, she could be replaced at any point by a rightful male heir. The makes it discriminatory.
The Australian monarchy does not violate sexual discrimination laws. Except for marriage equality, LGBTQ couples have many of the same rights as heterosexual couples in Australia. A human rights act was passed to ban discrimination against LGBTQ people in 1994 and sexual identity, gender identity, and intersex status were added to the protection in 2013.
If monarchies did not have equal power among the king or queen then they would violate sexual discrimination laws. This is not the case, for example the British monarchy have equal power for both the king and queen. In no other way can I see monarchies as sexist or in violation of rights.
The monarch does not violate sexual discrimination laws because the Queen is the head of everything. Although she does not do much, having a woman as a leader paves the way for both men and women to become a leader. As the queen, she has allowed both men and women to be in charge of different government departments of Australia.