public partial class Form1 : Form
    {
        Serialization_Class serialization_Class;
        Client client;

        public Form1()
        {
            InitializeComponent();
            tbx_countShoes.Text = "0";
            serialization_Class = new Serialization_Class();
            serialization_Class.ShoesList = new List<string>();
        }

        // кнопка "Добавить магазин"
        private void btn_addShoes_Click(object sender, EventArgs e)
        {
            if (tbx_shoes.Text != String.Empty)
            {             
                serialization_Class.ShoesList.Add(tbx_shoes.Text);                 
                int countShoes = Convert.ToInt32(tbx_countShoes.Text);
                countShoes++;
                tbx_countShoes.Text = countShoes.ToString();
                tbx_shoes.Text = String.Empty;
            }
        }

        // кнопка "Добавить клиента"
        private void btn_addClient_Click(object sender, EventArgs e)
        {
            serialization_Class.Name = tbx_name.Text;
            serialization_Class.Surname = tbx_surname.Text;
            serialization_Class.Age = Int32.Parse(tbx_age.Text);
            serialization_Class.Phone = tbx_phone.Text;

            client = new Client();
            client.ListClients.Add(serialization_Class);      // Здесь возникает ошибка !!!

            serialization_Class = null;
            tbx_countClients.Text = client.ListClients.Count.ToString();
            client.ListClients.Clear();
        }
    }

public class Client
    {
        private List<Serialization_Class> listClients;

        public List<Serialization_Class> ListClients
        {
            get { return listClients; }
            set { listClients = value; }
        }

    }

public class Serialization_Class
    {
        public Serialization_Class()
        { }

        private string name;
        private string surname;
        private int age;
        private string phone;
        private List<string> shoesList;

        public string Name
        {
            get { return name; }
            set { name = value; }
        }

        public string Surname
        {
            get { return surname; }
            set { surname = value; }
        }

        public int Age
        {
            get { return age; }
            set { age = value; }
        }

        public string Phone
        {
            get { return phone; }
            set { phone = value; }
        }

        public List<string> ShoesList
        {
            get { return shoesList; }
            set { shoesList = value; }
        }
    }