Hi:
You are correct. The embedded W8.1 product key in the BIOS should have automatically activated W8.1.
I have zipped up and attached a free utility below, that you can run, and it will provide the W8.1 product key in the BIOS.
If there is no product key, then HP installed the wrong motherboard in your PC (one without the W8.1 product key).
Some model x360-11 variations come with FreeDOS and those motherboards would not have product keys in the BIOS.
If that is the case, you will have to contact HP technical support for the country you live in and report the problem to them.
Hopefully, they will issue you a W8.1 OEM product key you can use to activate windows.
You can see from the service manual below, chapter 3, pages 16-17, that there are system boards with or without windows product keys.
http://h10032.www1.hp.com/ctg/Manual/c04644012