Redirect stderr to terminal and also file in bash

Yes, the tee command allows you to direct output to one or more files, as well as stdout.

As you pointed out in a comment, this doesn't work by itself because of the exec command.

This should do what you want:

exec 3>&1 1>test.log
echo "Check if this line is going to test.log"
exec 1>&3 3>&-
echo "Maybe this should go to stderr" | tee -a test.log >> /dev/stderr

I got the information about the way to restore stderr by properly setting up the original exec, from here, and combined it with tee.