Are Assignment Rules affected by the user type?

Yes, they are. You can set up assignment rules for different modules based on the user type.
Here are two conditions you must be aware of:

• If a group member is a Light Agent user, then you can include them as a member in the assignment rule for Leads and Contacts only. 
• If a group member is a Single App user, then you can add them to the assignment rule for the modules belonging to the apps the user has access to.

To know more about Users, click here.
To know more about Assignment Rules, click here.