IEnumerable<Entity> matchingEntities = entities.Where(e => e.Names.Any(n => myEntity.Names.Any(entityName => entityName.CompareNameObjects(n))));

public class Entity
{
   public string EntityIdentifier { get; set; }
   public List<Name> Names { get; set; }
}

public class Name
{
    public string FullName { get; set; }
    public string NameType { get; set; }

    public bool CompareNameObjects(Name name2)
    {
        return FullName == name2.FullName &&
               NameType == name2.NameType;
    }
}