Looking for the same objectname and then compare the structure

I am assuming that we are using List <T>[^] to implement your list of objects.

If you read the documentation of the List<T>.Contains Method[^] and IEquatable<T>.Equals Method[^] you will see that your statement will work if all the objects stored in the list implement the IEquatable interface.

Define the Equals function in the objects like:

public override bool Equals(Object obj)

with this implementation you can return false when the objects are of different types, see the sample in the interface documentation.

Do you control the class of which the objects are instances? If so, you can override Equals and == so that a == b returns what you want. E.g, a trivial example

class Test {
 int a; double b;

 public static operator bool == (Test a, Test b) { return (object)a == null ? (object)b == null : a.Equals(b); }
 public static operator bool != (Test a, Test b) { return !(a == null); }

 public override bool Equals(object o){
  if(o is Test){
   Test test = (Test)o;
   return a == test.a && b == test.b;
  } else return base.Equals(o);
 }

 public override int GetHashCode(){ return a.GetHashCode() ^ b.GetHashCode(); }
}

If not, and you don't know in advance what type the objects will be, you'll need to use reflection, make a sorted list of their fields/properties/etc (whatever you count as 'structure') and compare those.