Some landline and office phones are able to display text in addition to the calling number as the caller ID. Calls made with Twilio can only display a verified phone number or a number purchased from Twilio as the caller ID, but there are steps you can take so that your business name appears as the caller ID on some phones.
To display name information instead of a phone number when making outbound calls, a number must be registered in a central database. In the United States, this database is CNAM. (This is to display your name when calling new devices that have CNAM database lookup enabled; if your number is saved to a device's local address book, you don't need the CNAM registry).
Currently, it's not possible to register phone numbers purchased from Twilio in CNAM. However, if it is very important for you to display a name in your called party's caller ID, add another registered number as a validated caller ID in your Twilio project. Then, when you make outbound calls, use that caller ID as the "From" number. The name will display as if you're calling from that caller ID number.
Caller ID may or may not be rendered properly when a call is made across international borders. Please read our article on international caller id to learn more.