This always shows up a few minutes after I log in:
I have no idea what it is. If I click Report problem, it asks for my password:
This is exactly what I would do if I wanted to steal someone's password.
Why should I trust it, and what is it going to do next if I comply?
You're absolutely right, there is a possibility that this is some other software masquerading as Apport (the bug reporting software).
I wouldn't worry about it though, as this is the expected behaviour of Apport. If you have it enabled and a program crashes, a window similar to this will appear. Sometimes Apport needs to run as root to gather system information and send it to Launchpad.
If you don't want to run this risk, just disable Apport.
If you do decide to trust this prompt, you'll be able to inspect the report and find out which program is crashing on login.
To disable Apport:
gksu gedit /etc/default/apport and press Enter.enabled=1 with enabled=0.
