Update Tkinter UI from another thread

You can use queue.SimpleQueue to pass data from Server to MainPage so that MainPage can insert the data into the Listbox:

main.py:

from queue import SimpleQueue
...
class App(tk.Tk):
    def __init__(self, queue, *args, **kwargs):
        tk.Tk.__init__(self, *args, **kwargs)
        self.queue = queue
        ...

if __name__ == "__main__":
    queue = SimpleQueue()
    server = Server(queue) # pass queue to Server
    app = App(queue) # pass queue to App
    ...

server.py:

class Server:
    ...
    def __init__(self, queue=None):
        ...
        self.queue = queue

    def __accept_incoming_connection(self):
        while self.__is_running:
            client, client_address = self.__server.accept()
            # I want to insert text into listbox of the MainPage (msg_list) here
            if self.queue:
                self.queue.put(client_address) # or whatever information you want
    ...

main_page.py:

class MainPage(tk.Frame):
    def __init__(self, parent, controller):
        ....
        # call a monitor task periodically
        self.monitor_queue()

    def monitor_queue(self):
        if not self.controller.queue.empty():
            caddr = self.controller.queue.get()
            self.msg_list.insert(tk.END, str(caddr))
            self.msg_list.see(tk.END)
        self.after(100, self.monitor_queue)

Below is an example using similar structure as your code:

import tkinter as tk
from queue import SimpleQueue
from socket import socket, AF_INET, SOCK_STREAM
import threading

HOST = ""
PORT = 3000
ADDRESS = (HOST, PORT)

class Server(threading.Thread):
   __instance = None

   @staticmethod
   def get_instance():
      if Server.__instance is None:
         Server()
      return Server.__instance

   def __init__(self, queue=None):
      if Server.__instance:
         raise Exception("Server is singleton class")
      super().__init__()
      Server.__instance = self
      self._server = socket(AF_INET, SOCK_STREAM)
      self._server.bind(ADDRESS)
      self._server.listen()

      self.queue = queue

   def set_queue(self, queue):
      self.queue = queue

   def run(self):
      print("Server started")
      while True:
         try:
            client, client_address = self._server.accept()
            if self.queue:
               self.queue.put(client_address)
         except Exception as e:
            print(e)
            break
      print("Server terminated")

   def terminate(self):
      self._server.close() # self._server.accept() will raise exception and break the while loop

class SetupPage(tk.Frame):
   def __init__(self, parent, controller):
      super().__init__(parent)
      self.controller = controller
      tk.Label(self, text="Setup Page").pack(padx=100, pady=50)
      tk.Button(self, text="Goto Main page", command=lambda: self.controller.show_frame("MainPage")).pack()

class MainPage(tk.Frame):
   def __init__(self, parent, controller):
      super().__init__(parent)
      self.controller = controller

      self.scrollbar = tk.Scrollbar(self)
      self.msg_list = tk.Listbox(self, width=50, height=15, yscrollcommand=self.scrollbar.set)

      self.scrollbar.pack(side=tk.RIGHT, fill=tk.Y)
      self.msg_list.pack(side=tk.LEFT, fill=tk.BOTH, expand=True)

      self.monitor_queue()

   def monitor_queue(self):
      if not self.controller.queue.empty():
         caddr = self.controller.queue.get()
         self.msg_list.insert(tk.END, str(caddr))
         self.msg_list.see(tk.END)
      self.after(100, self.monitor_queue)

class App(tk.Tk):
   def __init__(self, queue):
      super().__init__()
      self.queue = queue

      container = tk.Frame(self)
      container.pack(side=tk.TOP, fill=tk.BOTH, expand=True)
      container.grid_rowconfigure(0, weight=1)
      container.grid_columnconfigure(0, weight=1)

      self.frames = {}
      for F in (SetupPage, MainPage):
         frame = F(parent=container, controller=self)
         frame.grid(row=0, column=0, sticky="nsew")
         self.frames[F.__name__] = frame

      self.show_frame("SetupPage")

   def show_frame(self, page_name):
      frame = self.frames[page_name]
      frame.tkraise()

if __name__ == "__main__":
   queue = SimpleQueue()
   server = Server(queue)
   server.start()
   app = App(queue)
   app.title("Server")
   app.geometry("600x450")
   app.mainloop()
   server.terminate()