Крестики-нолики (ввод - вывод)

Решение довольно простое, просто все обернуть в цикл и сделать ожидание нажатие ENTER от пользователя после того как он закончил игру и начинать цикл заново. Это все будет повторяться до того момента пока пользователь не закроете программу.

from random import randint
import os
# Отрисовка поля
# Старт игры, запрос ввода (если валидация == False, exception)
# Валидация ввода
# Переопределяем значения списка по индексу
# Проверка на победителя
# Переключаем игрока

while True:
    BOARD_SIZE = 3
    board = [i for i in range(9)]
    is_winner = False
    current_player = randint(0, 1)
    markers = {0: 'O', 1: 'X'}
    available_turns = (x for x in range(9))
    def draw_board():
        res = ""
        for i, v in enumerate(board):
            res += str(v) + " "
            if (i+1) % BOARD_SIZE == 0:
                 res += "\n"
        print(res)

    def validate(value = " "):
        if not value.isdigit() and int(value) not in available_turns:
            raise ValueError ("Enter valid value and try again")
        if board[int(value)] in ('X', 'O'):
            raise ValueError ("This value has already played")
        if '.' in value:
            raise ValueError('Number must be int')

    def check_winner():
        current_marker = markers[current_player]
        if board[0] == current_marker and board[4] == current_marker and board[8] == current_marker or \
                board[2] == current_marker and board[4] == current_marker and board[6] == current_marker or \
                board[0] == current_marker and board[1] == current_marker and board[2] == current_marker or \
                board[3] == current_marker and board[4] == current_marker and board[5] == current_marker or \
                board[6] == current_marker and board[7] == current_marker and board[8] == current_marker or \
                board[0] == current_marker and board[3] == current_marker and board[6] == current_marker or \
                board[1] == current_marker and board[4] == current_marker and board[7] == current_marker or \
                board[2] == current_marker and board[5] == current_marker and board[8] == current_marker:
            return True
        else:
            return False

    for i in range(9):
        try:
            input_error = True
            draw_board()
            while input_error:
                choice = input(f"Pllayer {markers[current_player]} enter your number Введите число:\n")
                validate(choice)
                input_error = False
            board[int(choice)] = markers[current_player]
            #проверить победителя
            is_winner = check_winner()
            if is_winner == True:
                print(f'Player {markers[current_player]} won the game!!')
                break
            current_player = 0 if current_player == 1 else 1
            # if is_winner == True:
        except ValueError as ex:
            print(ex)

    if is_winner == False:
        print('Draw')
    input('Хотите сыграть еще раз? Нажмите ENTER')
    os.system('cls')

Переместите цикл в функцию и вызывайте ее в бесконечном цикле с выходом по условию (если игрок ответит Нет)... Как-то так, например:

from random import randint


BOARD_SIZE = 3

# Отрисовка поля
# Старт игры, запрос ввода (если валидация == False, exception)
# Валидация ввода
# Переопределяем значения списка по индексу
# Проверка на победителя
# Переключаем игрока

def draw_board():
    res = ""
    for i, v in enumerate(board):
        res += str(v) + " "
        if (i+1) % BOARD_SIZE == 0:
             res += "\n"
    print(res)

def validate(value = " "):
    if not value.isdigit() and int(value) not in available_turns:
        raise ValueError ("Enter valid value and try again")
    if board[int(value)] in ('X', 'O'):
        raise ValueError ("This value has already played")
    if '.' in value:
        raise ValueError('Number must be int')

def check_winner():
    current_marker = markers[current_player]
    if board[0] == current_marker and board[4] == current_marker and board[8] == current_marker or \
            board[2] == current_marker and board[4] == current_marker and board[6] == current_marker or \
            board[0] == current_marker and board[1] == current_marker and board[2] == current_marker or \
            board[3] == current_marker and board[4] == current_marker and board[5] == current_marker or \
            board[6] == current_marker and board[7] == current_marker and board[8] == current_marker or \
            board[0] == current_marker and board[3] == current_marker and board[6] == current_marker or \
            board[1] == current_marker and board[4] == current_marker and board[7] == current_marker or \
            board[2] == current_marker and board[5] == current_marker and board[8] == current_marker:
        return True
    else:
        return False


def game(is_winner, current_player, ):
    for i in range(9):
        try:
            input_error = True
            draw_board()
            while input_error:
                choice = input(f"Player {markers[current_player]} enter your number Введите число:\n")
                validate(choice)
                input_error = False
            board[int(choice)] = markers[current_player]
            #проверить победителя
            is_winner = check_winner()
            if is_winner == True:
                print(f'Player {markers[current_player]} won the game!!')
                break
            current_player = 0 if current_player == 1 else 1
            # if is_winner == True:
        except ValueError as ex:
            print(ex)

    if is_winner == False:
        print('Draw')

new_game = True
while new_game:
    print('Новая игра')
    board = [i for i in range(9)]
    is_winner = False
    current_player = randint(0, 1)
    markers = {0: 'O', 1: 'X'}
    available_turns = (x for x in range(9))


    game(is_winner, current_player, )
    new_game = int(input('Еще игру? (1 - да, 0 - нет)'))