are govs required to comply with GDPR and data breaches laws?

Yes, but unelected bureaucrats only impose fines on the private sector.

what would be the point of the government fining itself though?

Now that I'm thinking of it, it would create the need for an extra gaggle of bureaucrats to oversee the process,so I suppose someone might see a point to it ...

You may think you're funny or something, but boy do I have news for you.

There absolutely are fines for French administrations. And, knowing the French tax system, they've probably found a way to levy VAT and some other taxes on top of those fines.

Do you mean fines for tiny companies?

There are carve-outs to allow for governments to make exceptions, but it's besides the point.

If the government were to hold themselves to account, they would fine themselves some amount N, and pay itself N using your taxes. It also wastes other finite resources for all the paperwork and legal action involved that could be used for something else.

Speaking pragmatically, there's no point trying to hold the government itself to it's own laws. The only time citizens do hold the government accountable, it's always done in the form of hangings, or the guillotine in France's case.