Public Mobile does not offer the option of outgoing name display like Telus & Koodo *postpaid* accounts do. If your line is in fact with Public Mobile and the wrong name is showing up on caller ID, it's either because the person you're calling has the number stored in their contacts under that name, or more likely their phone is using a third party service to lookup the name associated with that number and displaying it for the incoming call.
My Pixel phones do this, for example. When the caller ID number comes in the phone reaches out to the interwebs to see if the number is associated with a known name or business. Even for outgoing calls my phone will often do this if I'm calling a business (i.e. when I look in my call logs I'll see the business name, even though it's not in my contacts).
You'd probably have to find these third party services and try to convince them to update their records, but I have no idea how that works or whether you'll have any luck.
