That costs $$$$ usually.
You're essentially asking for a list of all companies, incorporated or not in a country - US I'm assuming?
Does that include every 7-11 and McDonalds, franchises separate from HQ? In every State and County?
SEC has this list, there will be others but usually we buy the data from a company called Dunn & Bradstreet. If you're affiliated with an academic institution I'd ask the librarian or check and see if you had access to WRDS or something similar for the US.
https://www.sec.gov/rules/other/4-460list.htm
Regarding classification, it depends on what you're trying to do....description will be text data so that will be an interesting exercise. The Dunn & Bradstreet data will have the NAICS code and many other business demographic data that help for clustering.
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