I'm a non-native speaker . I noticed that there are two kinds of statement to describe polity , federal republic of Germany and commonwealth of Austrilia . So I'm confusing what's the difference between federal republic and commonwealth?
The obvious answer is that Australia is not a republic: it still has a Queen. Consequently it isn't called a Federal Republic. The country does have a Federal construction, with a number of self-governing states and territories [for example, a car will need to be re-registered if transferred between states], and the Federal capital in its own territory outside the other states. This structure mirrors the United States of America.
In the case of Australia, the structure is historical as the states/territories were originally separate British colonies, becoming a federated colony which they called a Commonwealth on 1 January 1901 and independent in 1942 (backdated to 1939). Each separate colony retained a degree of independence from its neighbours and the whole.
The word commonwealth means of or for the common good (cf weal).
weal noun [mass noun]
that which is best for someone or something:
I am holding this trial behind closed doors in the public weal
Federal is an adjective, so had it been decided in 1901 or 1942 to use that word, it would have needed a noun to go with it, and Republic is obviously inappropriate. The word Commonwealth, with the constituent common, does indicate some sort of commonality between separate entities and can thus indicate a federation without requiring either an additional noun or a separate “federal” adjective.