I am having the exact same problems on clean image Raspbian install on RPi3b+. I've tried new SD cards, a new power adapter, and even a new RPi3b+, without any luck. The install script seems to set DNS 127.0.0.1 "too early", and the rest of the script fails. If I set DNS to i.e. 8.8.8.8, I can now retry the install script, and it will succeed. But the Pi-hole installation will mess up the Raspbian system, eth0 has lots of problems, and in my case it always ends in system crash (kernel panic?).
I created a new post explaining my problems: [RESOLVED] Endless problems with Pi-hole after Raspbian 2020-02-05 update
And yes, this started immediately after the Feb 2020 Raspbian update.