The short version:
Is there any way to set up automatic public-key-based ssh authentication from one Linux account to two different Github accounts?>
You need to create two sets of (public/private) keys, one for each account.
You can reference them through an ssh config file, as detailed in "GitHub: Multiple account setup"/
#Account one
Host github.com
HostName github.com
PreferredAuthentications publickey
IdentityFile /c/Users/yourname/.ssh/id_rsa
User git
#Account two
Host ac2.github.com
HostName github.com
PreferredAuthentications publickey
IdentityFile /c/Users/yourname/.ssh/id_rsa_ac2
User git