EZ Exporter can export only the internal Shopify auto-generated company ID and location ID.  It's currently not possible to export the user-specified company ID and location ID.

To export these IDs, you can enter the following in the Custom Fields or Calculated Fields section:

• {{ company.id }}
• {{ company.location_id }}

These IDs correspond to the ID shown in the URL in the Shopify Admin:

You could potentially use a key/value mapping based on those auto-generated IDs to map them to the actual names in the Calculated Fields section using the get_attribute_value() function as a workaround.  However, the downside of this approach is you'll have to keep updating the formula as you add new company records.