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There are two issues here: licensing (whether you can ride) and registration (whether you can transfer a vehicle to your name).
You can use your overseas driver's licence to drive in New South Wales as long as you are a temporary visitor. If you wish, you can obtain a NSW temporary visitor licence.
If you stop being a temporary visitor (e.g. if you get a permanent visa), then you have 3 months to obtain a local driver's licence.
As a Polish licence holder, you can obtain a NSW driver's licence without needing to pass the knowledge and driving tests if you are over 25 years of age.
Anyone over 16 years of age can register a vehicle. However, it is useful to have a NSW driver's licence as it will make the process easier (e.g. you will be able to transfer the registration online).