You can see and access your regular saved contacts through CaptionMate by allowing contact permission in Settings. If you aren't seeing the names of saved contacts appear when you receive calls through CaptionMate, that means the app's permission to access your contacts is off.
How to turn on contact permission on iPhone:
Open CaptionMate
Tap Settings
Tap Permissions; it is listed lower on the page under the 'Technical' subheading.
Tap the button next to Access to contacts. (This should be at the top of the menu)
Tap Contacts
Select Full Access
Tap Allow Full Access to confirm
How to turn on contact permission on Android:
Open CaptionMate
Tap Settings
Tap Permissions; it is listed lower on the page under the 'Technical' subheading.
Tap the button next to Access to notifications. (This should be at the top of the menu)
Tap Continue to proceed to your Settings app
Tap Permissions
If 'Contacts' is under not allowed, tap it and select Allow