To be more precise, from what I understand. You upgrade your license (the OEM SLP one or the one on your sticker, which are technically two different licenses. Draw your own conclusion from that and how you can abuse this). During the upgrade process, you get a new product key. This product key, from what I've seen, is the same for every machine that is upgraded. That Win10 product key, for Home, ends with 8HVX7, for Pro, ends with 3V66T. Google that if you want.
What really happens is that a hardware hash is sent to Microsoft during the upgrade process. This hardware hash allows you to use those generic keys in the future (well, depending whether you had Home or Pro... Obviously), which means you can just use the generic ISO Microsoft provides (Finally, an official re-installation ISO! I've been waiting years for that). You can not use those generic keys on non-hashed hardware (Yes, I tried to see what happens). It will not activate.
However, your 7 license will remain fully functional. At least, that's my experience.
What would be an interesting test would be the following: Install Windows 7 in a VM, clone it, but don't run the second instance. Start the first instance, upgrade to 10. Keep it on 10. Now launch the second instance, which is 7 and never upgrade it. See if both remain active. This definitely violates the Microsoft licenses you have, but it would be interesting to see what happens. My prediction: both stay activated, but I'm not sure. I haven't tested it.