Whenever you make FaceTime audio or video calls from your iPhone, Apple automatically uses your phone number or Apple ID email address as the caller identification. So when someone that you're calling sees the incoming call, they'll see it's from your phone number. But what if you'd rather it be a different identifier?
If your phone number or email address is already in the person's contact list, you'll likely only show up as "[Your Name] would like FaceTime" on their Mac or iOS device, so changing the caller ID in this instance isn't really necessary. However, if you're FaceTime calling someone fresh, they will instead see either your phone number or email address instead of your name.
If your phone number has an odd area code or your email address isn't very indicative that you're the one asking for a FaceTime audio or video call, you may want to change the caller ID to an email address with your name in it so recipients won't reject your calls.
Also, if you have an iPhone for work and an iPad for home use only, you will probably want to make sure that your work iPhone is using your work phone number or email address and that your iPad at home is not.
On your iPhone, open up the Settings app, then tap on "FaceTime" to view its available preferences. In the FaceTime settings, you'll see a section called Caller ID, and only one from the list will be check marked. This is likely your iPhone's phone number, but could also be your Apple ID email address if you've never set up a phone number on the device.
All you have to do is tap one of the other emails in the list to make it show up whenever you make a FaceTime call. As long as it's check-marked, you're good to go.
You'll still be able to be reached by any of the check-marked items in the You Can Be Reached at FaceTime At list above it, only whenever you make a FaceTime call, they will see your Called ID choice, either on their screen right away when you call or in their call history in the Phone app (if they have you listed as a contact).
If you'd like to add an email address for your Caller ID that you don't see in the list of selectable emails, you can learn how to add emails to FaceTime here.