Why does a DHCP-provided DNS IP address becomes a static one in Windows 10?
Solution 1:
While typing this question I got the answer from someone who had the same problem: the Obtain DNS server address automatically setting is "inherited", in the sense that if any interface on the machine is set to have a static DNS server, the others will switch to a static one upon receiving the DHCP information.
The solution is therefore to look though all the interfaces on the machine and make sure that all DNS settings are set to DHCP.