An Assigned To field points to people and groups, not to another list. I see 3 ways to address your issue:
- Instead of list B, create a SharePoint group, and have your Assigned To field point to that group.
- Instead of an Assigned To field, use a lookup field that points to list B
- A combination of both. For example use a lookup field that points to list B, then have a workflow copy the lookup value to the Assigned To field.
The choice will depend on your specific needs and constraints. The first solution is the easiest if it works in your case.
For the third option, use the e-mail address as key: copy the e-mail address of list B to the Assigned To field, SharePoint will find the related people.