How very American that the requirement to register is to obtain a private fee for service business identification, not some kind of institutionalised public interest registry.
Yes. It's totally bizarre that a formalised business identity is held in a commercial enterprise, not some kind of not for profit mutuality or a state enterprise.
Who requires them to do it isn't the point, what Google decided is the formalism to meet EU requirements is the point.
Here is a european collated list of worldwide business registries. The Australian one is a gov.au. the US one is the SEC not D&B
The SEC is not a registry of all businesses in the US. The SEC concerns itself with business that sell shares to the public; this would not cover (for example) a single-member LLC that a software developer might use.
In the US, businesses are chartered and registered by the states, not the Federal government. There is no Federal equivalent to the UK's Companies House.