There is much more to be done in order to get Maya to work on Ubuntu: installing libpng1.5.15 is only one out of many steps.
The full process is described here: https://knowledge.autodesk.com/support/maya/learn-explore/caas/simplecontent/content/installing-maya-2020-ubuntu.html
To solve the libpng problem, simply download libpng1.5.15 from the libpng project: https://sourceforge.net/projects/libpng/files/libpng15/older-releases/1.5.15/, and install it to /usr (where Maya expects it to be. You can also let it install to the default /usr/local and set up LD_LIBRARY_PATH instead). It can perfectly fine coexist with modern versions of libpng.
To build it, make sure autotools and libtool is installed, and do:
tar xvzf libpng-1.5.15.tar.gz
sudo make install
Now the Maya installer will run, and then complain about not supporting the operating system and exit. The full process described on the AutoDesk Knowledge Network linked above has 14 steps. By following them exactly, I just got it to work.