There are also many places in the US where influenza shots aren't contractual, they're law, particularly for those in schools, long-term care facilities, and healthcare settings. And vaccine mandates have hundreds of years of precedent in the US. It ain't the mandates that are new, it's the political climate.
These mandates aren't only occupational. You may be required by law to be vaccinated to attend school in some places, or to be in certain care facilities.
And the CDC wrote zero of these laws. The CDC's rephrasing of an FAQ article on their website may be a good starter for a right wing conspiracy theory article, but it has no bearing on this topic.
Laws apply to humans, who read the CDC website for both health information and guidance on compliance. When the US declared a Covid emergency in 2020, specific laws were invoked, some of which require policy input from HHS, including the CDC. Government agencies routinely interpret laws for application in specific circumstances. Such agency interpretations carry legal weight and real-world consequences. CDC definitions are not changed often, or casually.