Because I have not enough reputation points on stackoverflow to provide the proper answer, but can answer here. Here we go.
The problem itself lies in the fact that after an update/upgrade, the kernel version Ubuntu picks is a generic one, while the Hyper-V VM now expects there to be an Azure Linux kernel.
To fix this, you will notice you can switch to the terminal (as mentioned in the other answer) via Control + Alt + F3.
Log in, do a sudo apt update && sudo apt upgrade, but make sure you do not do an apt autoremove so you do not accidentally remove potential required previous kernels you might need.
After updating, you will have the latest generic ubuntu kernel as the active kernel in grub that will be booted up, which will cause the darkgray/black screen with hanging cursor issue if you reboot.
Now, perform a sudo reboot and switch to the terminal again with Control + Alt + F3.
Open /etc/default/grub with your favorite text editor (vi/vim/nano) with sudo and make the following changes:
Comment out and add the lines like this:
#GRUB_DEFAULT=0
GRUB_DEFAULT=saved
#GRUB_TIMEOUT=0
GRUB_TIMEOUT=20
GRUB_SAVEDEFAULT=true
Save the file and close the editor.
(nano: Control + O, Control + X)
(vi: Escape, :, W, X)
The option you activated by choosing saved, means the kernel you are going to select will be remembered over reboots.
Update the grub list, execute: sudo update-grub
Reboot again, hold shift on boot or press escape during the boot to open the grub bootloader and choose the most up-to-date version of the Linux-5.15.x-xxxx-azure kernel.
After doing this, the system works again.
If you want to set the timeout to 0 again, reopen the grub file and change which of the GRUB_TIMEOUT lines you commented out, then perform the sudo update-grub again.