Как остановить дополнительный поток нажатием кнопки?

Если я вас правильно понял, вам надо добавить проверку, например:

            if not self.stop:
                continue

Да, я немного упорядочил ваш модуль main.py.
Поменяйте импорты и раскомментируйте строки, связанные с webdriver.

main.py

import sys
'''
from PySide6.QtWidgets import *
'''
from PyQt5 import QtCore, QtGui, QtWidgets
from PyQt5.Qt import *

#from Controller.Potok_Controller import Potok_Controller
from q1436700_Potok_Controller import Potok_Controller

#from View.ui_potok import Ui_MainWindow
class Ui_MainWindow(object):
    def setupUi(self, MainWindow):
        if not MainWindow.objectName():
            MainWindow.setObjectName(u"MainWindow")
        MainWindow.resize(400, 300)
        self.centralwidget = QWidget(MainWindow)
        self.centralwidget.setObjectName(u"centralwidget")
        self.verticalLayout = QVBoxLayout(self.centralwidget)
        self.verticalLayout.setObjectName(u"verticalLayout")
        self.frame = QFrame(self.centralwidget)
        self.frame.setObjectName(u"frame")
        self.frame.setFrameShape(QFrame.StyledPanel)
        self.frame.setFrameShadow(QFrame.Raised)
        
        self.verticalLayout_2 = QVBoxLayout(self.frame)
        self.verticalLayout_2.setObjectName(u"verticalLayout_2")
        self.plainTextEdit = QPlainTextEdit(self.frame)
        self.plainTextEdit.setObjectName(u"plainTextEdit")
        self.verticalLayout_2.addWidget(self.plainTextEdit)
        self.verticalLayout.addWidget(self.frame)

        self.frame_2 = QFrame(self.centralwidget)
        self.frame_2.setObjectName(u"frame_2")
        self.frame_2.setFrameShape(QFrame.StyledPanel)
        self.frame_2.setFrameShadow(QFrame.Raised)
        self.verticalLayout_3 = QVBoxLayout(self.frame_2)
        self.verticalLayout_3.setObjectName(u"verticalLayout_3")
        self.pushButton = QPushButton(self.frame_2)
        self.pushButton.setObjectName(u"pushButton")
        self.verticalLayout_3.addWidget(self.pushButton)
        self.pushButton_2 = QPushButton(self.frame_2)
        self.pushButton_2.setObjectName(u"pushButton_2")
        self.verticalLayout_3.addWidget(self.pushButton_2)
        self.verticalLayout.addWidget(self.frame_2)

        MainWindow.setCentralWidget(self.centralwidget)
        self.retranslateUi(MainWindow)
        QMetaObject.connectSlotsByName(MainWindow)

    def retranslateUi(self, MainWindow):
        MainWindow.setWindowTitle(QCoreApplication.translate("MainWindow", u"MainWindow", None))
        self.pushButton.setText(QCoreApplication.translate("MainWindow", u"Start Thread", None))
        self.pushButton_2.setText(QCoreApplication.translate("MainWindow", u"Stop Thread", None))


class MainWindow(QMainWindow, Ui_MainWindow):
    def __init__(self, parent=None):
        super(MainWindow, self).__init__(parent)
        self.setupUi(self)
        
        self.pushButton.clicked.connect(self.open_window)
        self.pushButton_2.clicked.connect(self.stop)
        self.pushButton_2.setEnabled(False)
        
        self.myThread = Potok_Controller()
        self.myThread.mysignal.connect(
            lambda text: self.plainTextEdit.appendPlainText(text))
        self.myThread.finish_signal.connect(lambda: print('finish_signal'))

    def open_window(self):
        self.myThread.stop = True
        self.pushButton.setEnabled(False)
        self.pushButton_2.setEnabled(True)
        self.myThread.start()
        
    def stop(self):
        self.myThread.stop = False
        self.pushButton.setEnabled(True)
        self.pushButton_2.setEnabled(False)
        
        
if __name__ == '__main__':
    app = QApplication(sys.argv)
    window = MainWindow()
    window.show()
    sys.exit(app.exec())

q1436700_Potok_Controller.py

# ???import time

#from selenium import webdriver
'''
from PySide6.QtCore import QThread, Signal                          # PySide6
'''
from PyQt5 import QtCore, QtGui, QtWidgets                          # PyQt5
from PyQt5.Qt import *                                              # PyQt5


class Potok_Controller(QThread):
    '''
    mysignal = Signal(str)                                          # PySide6
    finish_signal = Signal()                                        # PySide6
    '''
    mysignal = pyqtSignal(str)                                      # PyQt5
    finish_signal = pyqtSignal()                                    # PyQt5
    
    def __init__(self, parent=None):
        super(Potok_Controller, self).__init__(parent)
        self.url = ['https://google.com/', 'https://youtube.com/', ]
        self.stop = True

    def txt(self):
        while self.stop == True:

            for url in self.url:
            
# +++ vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv            
                if not self.stop:
                    continue
# +++ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

                self.mysignal.emit(f'open {url}')
#                chrome = webdriver.Chrome(executable_path='chromedriver\\chromedriver.exe')
#                chrome.get(url=f"{url}")

# ???                time.sleep(10)
                self.msleep(10 * 1000)
                
#                chrome.close()
#                chrome.quit()
                #chrome.save_screenshot('datadome_regular_webdriver.png')
                
        self.finish_signal.emit()

    def run(self):
        self.txt()