Login box on one site to log you into another site too
Sure it's possible. There are many ways to do this.

A simple solution, if both sites are running the same drupal version, would be to make a custom form, and connect to the other sites database, and get the login form. You would need to make sure that the submit path would go to the project manage site.

When the form is submitted, they would automatically go the that site, and if user/pass is correct, they would also be logged in.

Disclaimer: I haven't actually tried this before, but it should work just fine. Drupal wont know where the form was submitted, so should be the same as being on their project site and using the login form.