The contents of that green bar come from the certificate itself, which you will have to get reissued to change it. The certificate's O attribute in the subject (organization), along with the C attribute (country) determine what is displayed. If they are absent, it will simply display the primary subject domain name from the certificate.
Also, the name (and usually the green bar) is only displayed for EV (extended validation) certificates. If you don't have one, you would normally have just the domain name, or no information bar at all. How this is displayed is of course dependent on the browser; recent versions of firefox display your domain name on a teal background in the address bar for non-EV certificates, whereas chrome just displays a lock icon and formats "https://" in green type.
You can't get a wildcard EV certificate, so to get this functionality in most browsers you would have to see if they would issue you an EV certificate with each of your subdomains as alternate names (the one that signs verisign.com is like this).
Verifying your company name and number, and usually inspecting your personal ID, incorporation, and a document giving you authority to act on behalf of the corporation are all expected parts of getting an SSL certificate issued in your company name at all.