With iOS 13 (the latest iPhone software) you can control what information other people see when you message them. For example, you can choose to share your Apple ID photograph, your name etc. From memory, this is something that your phone will ask you to set when you setup your phone. It won’t actually share your Apple ID though.
An Apple ID will be shown instead of your phone number if you select it under "Start New Conversations From" in iMessage settings. This also occurs for iMessage users who do not have a phone number because they don't have an iPhone.
“Apple ID” meaning email address. As konqerror pointed out, you can choose whether to use the Apple ID’s primary email address as the ‘handle’ for iMessage/FaceTime instead of the phone number. You can add more email addresses as well if you want to conceal your primary email address.