GV does not provide Caller Name, only Caller Number.
Probably the names you saw with the old phone had been programmed into the phone along their numbers. It then used the Caller Number from GV to look into its store to find and display a name.
When you took the new phone to another house the phone service there did provide Caller Name and the phone displayed it.
So if the new phone can be programmed to do the same as the old phone you can do that. Or you can do as many others do and route your incoming GV calls through eg Callcentric which does provide Caller Name.