So in practice, ANAC conducts the training according to the CASA syllabus but CASA conducts the theory exams and flight tests and it is CASA that issues you a licence.
ANAC has its own Approved Testing Officer i.e CASA allows this person to conduct flight tests on its behalf.
ANAC is a Registered Training Organisation (RTO) approved by the Australian Federal Government to train overseas students:
Specifically by Australian Skills Quality Authority (ASQA)
ANAC is also an approved CAAC school (Civil Aviation Authority China) with authority to train Chinese cadet students.
In general, airlines and government departments do not recognise individual flying schools over others. Instead they recognise the qualification (which as we have said, is issued by CASA). There are no government flying schools in Australia.
Australian training is regarded as of a very good standard so your decision to train in Australia will be a good one.