List<Person> people = new List<Person>();
 people.Add(new Person() { Name = "Том", Id = "Id_1" });
 people.Add(new Person() { Name = "Том", Id = "Id_2" });
people.Add(new Person() { Name = "Ипполит", Id = "Id_3" });
people.Add(new Person() { Name = "Евкакий", Id = "Id_4" });
people.Add(new Person() { Name = "Евкакий", Id = "Id_5" });

foreach (var eee in people)
        {
            var key = eee.Name;
            if (!dictionaryElement.ContainsKey(key))
            {
                dictionaryElement[key] = new List<Id>();
                dictionaryElement[key].Add(eee.Id);
            }
        }

if (!dictionaryElement.ContainsKey(key))
    dictionaryElement[key] = new List<Id>();
dictionaryElement[key].Add(eee.Id);