I just solved the issue following this video. Using the command provided in solution 1 did the trick.
The command I used is the following:

sudo ln -s /usr/local/mysql/lib/libmysqlclient.21.dylib /usr/local/lib/libmysqlclient.21.dylib

The explanation of how this works is in that video, for anyone stepping with this problem in the future.
Thanks @jsulm for your help and time.

Hi, to get that missing the driver dll (ibmysqlclient.21.dylib) perhaps you have not installed MySQL on your Catalina PC? For more see here

Looks like you installed all your stuff in the base conda environment which is a bad idea.

I would recommend re-install conda cleanly and create a proper environment to install PySide2.