I indeed set my HA machine as the DNS server, or intended to do so by setting [my HA machine IP address as the main and secondary DNS server address on my WiFi router].
That explains why my test with AdGuard Home was unsuccessful.
I considered Nabu Casa solution and gave up when I realized it is not free.
I should have specified that I’m looking for a rather cost free solution.
In fact, it’s the opposite
When using my phone, as long as I’m not connected to my home WiFi remote access works perfectly (via the Companion app). However, once I’m connected to my home WiFi and tried to access HA via the Companion app, it shows the error can’t connect to Home Assistant.
I will reply later today with more details regarding my network configuration.