That does not reset the carrier name, it just seems to alter what is visible in, e.g., the Messages app, so you know which SIM is operating. (You can also change the colour, so that text in the phone history and phone screen colours change depending on SIM.
I have a dual-SIM P6a. The physical SIM correctly swaps from the network wholesaler to the VMNO carrier name, but the same is not true for the eSIM. On startup the network wholesaler name briefly appears before switching to the generic Your provider. This also happens when the physical SIM is not in use, i.e., when the eSIM is the only one operating.
This is intriguing, and somewhat amusing. But it would be nice to be able to fix it, too. Has anyone manged to find a way?