Unable to connect to the internet

Your debug log shows that your Fritzbox router is advertising as set of its own IPv6 addresses as DNS resolver:

*** [ DIAGNOSING ]: contents of /etc

-rw-r--r-- 1 root root 998 Nov 28 13:11 /etc/resolv.conf
   nameserver fd00::<redacted>70
   nameserver 2003:<redacted>70
   nameserver 2003:<redacted>87
   search fritz.box

I'd recommend to disable propagation of IPv6 DNS server addresses altogether.
See Unresolved ipv6 adress in my top list - #4 by Bucking_Horn for suggestions on how to achieve that with a Fritzbox.

In addition (and unrelated to your DNS observations), your webserver seems to be configured for a different IP (from the default range of your Fritzbox):

*** [ DIAGNOSING ]: contents of /etc/pihole
-rw-rw-r-- 1 1000 pihole 203 Nov 28 13:11 /etc/pihole/pihole-FTL.conf

You should adjust your container's FTLCONF_LOCAL_IPV4 for your new address.

In case your Fritzbox is still using a network, your would not be reachable from that network, unless routing is configured accordingly.

Your above nslookup may suggest that you are using separate subnets, which could explain your observation.

If you are indeed on a single subnet, something on the client issuing that nslookup may interfere with DNS operations.