Thanks @Black Mamba, but that would work only if I knew German...
Nonetheless, I was able to figure out the solution with the help of a very efficient lady from the Swisscom mobile support team, Roberta, who in two different calls was able to sort out the issue.
In a nutshell, it is NOT possible to change the master device from a plain SIM to an ESIM without scrapping the old plain SIM.
Roberta worked out a way to first disable the plain SIM on the former master phone, and then create an ESIM profile right on the My Swisscom page, where in a matter of minutes I found the QR code to scan.
I panicked for a few minutes when I noticed that the plain SIM had already been disabled, while the new ESIM profile QR code to scan had not yet appeared on the My Swisscom page... it took a few more minutes of waiting and the issue was finally sorted out.
So here are the steps I undertook:
1- I swapped the SIM card from the old iPhone 8 to the new iPhone 13 mini. Doing this made the iPhone 13 mini the master device, still using the old plain SIM, though.
2- then, in a call with Swisscom support, I had Roberta order a new ESIM profile for the same phone, which appeared like this in the My Swisscom portal:
3- after the ESIM activation process, in a few minutes here is what I found on the My Swisscom page:
4- after scanning the QR code, I could finally receive the confirmation call back from Roberta, who then proceeded to order and send me via post a new replacement plain SIM card, which is now active as secondary SIM on the same line, as I wanted to do in the first place.
This is my current situation on the cockpit, precisely like I wanted it to be:
Last point, in spite of the link to this Apple documentation that I found on the Swisscom mobile website, the procedure to create a new ESIM profile on the phone from the old plain SIM on the same phone did not work.
Here is the Apple documentation, useful but not applicable in my case, for whatever reason:
https://support.apple.com/en-sa/HT210655
This other useful Apple documentation talks about how to use the dual SIM feature on the iPhone:
https://support.apple.com/en-euro/HT209044
Hopefully this will be helpful to others!
Bye, Luca