

No, you don’t need to carry ID on you — many people that don’t drive (and many others that do) don’t have IDs.
You’ll also note that the state-issued ID must indicate the holder is a US-citizen. Most states have an upgraded version (enhanced) ID/ Drivers License that can be used to this end but it typically requires additional verification and an added fee — it is not provided by default. Non-citizens such as visa-holders and even non-legal residents are also able to obtain a state-issued ID, though not an enhanced drivers license.
The second part of that clause mentions accompanyment of proof of citizenship. This is usually a birth certificate which is often lost and costs money and takes roughly a month for states to tender a copy to the “holder”. Not sure of other states, but at least the few I am aware of do not provide a legal copy by default. Instead these must be ordered from the state.
bruh