Certificate of compliance in Canberra?

Ultimately the home owners responsibility, but usually done by good builders and not done by cheaper/dodgy builders.

It will cost you about $55 assuming your C of O has already issued.

You might be getting confused …
A Certificate of Occupancy or Use is given when you complete building works, such as a new house, an extension, new building etc. For every bit of building work you do, you receive a Certificate of Occupancy or Use.
A Compliance Certificate is a once-off certificate that the lessee is responsible for obtaining. A Compliance Certificate is a certificate that says you have satisfied the conditions in your Crown lease.
You don’t need to obtain your compliance certificate at the same time as when you develop your residence.
Usually people obtain their ccompliance certificate because the banks requests a copy. Otherwise, some people will develop their block, live there for 20 years, obtain many other Certificates of Occupancy or Use with other works that occur on the block, and gthen they obtain one when they decide to sell.

Is it a free standing single dwelling (house), or a unit/townhouse?

If it is a free standing dwelling you should have received your Certificate of Compliance and Certificate of Occupancy and Use at the completion of the building work.

If it is a unit/townhouse (i.e. part of a multi unit complex) and hasn’t been sold since construction you will need to apply for a Certificate of Compliance through ACTPLA or whatever it is called now. The loan provider for anyone purchasing the property will want to see this.

It would be interesting to discover if the builder has primary responsibility for providing the COE, or the certifier.

I would guess it depends on the contract with the builder in question. From my own single experience building a house the builder provided it. But we had a building manager who may have done that rather then the builder.
If you had a partially completed house which you were responsible for finishing, like additional wiring, plumbing or whatever then it would have been your responsibility to get the compliance certificate.

You as the lessee