Of course. When a call is received on the mobile device, the phone number is shown on the display. If the phone number is part of a contact (in the Contacts app), the name from the contact is displayed.
On the iPhone, Apple's contacts app is primarily used for this resolution (incoming call identification). However, there is also an interface in the phone app that allows resolution of the phone number by other apps. For example Webex or MDM solutions.
The suggestion is that Thunderbird also supports this interface with its contacts, so the phone numbers are resolved via the contacts, from Thunderbird. Then you would need only one solution, Thunderbird.