Theme in MaterialApp not working in widgets Flutter

This is because the color of your Container widget is taken from context, which is an argument of the build method of MyApp.

MaterialApp has a theme, but MyApp doesn't have a theme. So when you use the context of MyApp to get the primary color, that context doesn't have any theme yet and you get the default primary color.

Instead, what you can do is:

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Spotify UI',
      debugShowCheckedModeBanner: false,
      themeMode: ThemeMode.dark,
      darkTheme: ThemeData(
        // ...
      ),
      home: MyWidget(),
    );
  }
}

class MyWidget extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Column(
      children: [
        Expanded(
          child: Container(
            color: Theme.of(context).primaryColor,
          ),
        )
      ],
    );
  }
}

Your MaterialApps body yet does not contain updated BuildContext with themes in its body. Wrap your Column with Builder to provide updated context:

import 'package:flutter/material.dart';

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Spotify UI',
      debugShowCheckedModeBanner: false,
      themeMode: ThemeMode.dark,
      darkTheme: ThemeData(
        brightness: Brightness.dark,
        appBarTheme: const AppBarTheme(backgroundColor: Colors.black),
        scaffoldBackgroundColor: const Color(0xFF121212),
        backgroundColor: const Color(0xFF121212),
        primaryColor: Colors.black,
        accentColor: const Color(0xFF1DB954),
        iconTheme: const IconThemeData().copyWith(color: Colors.white),
        fontFamily: 'Montserrat',
        textTheme: TextTheme(
          headline2: const TextStyle(
            color: Colors.white,
            fontSize: 32.0,
            fontWeight: FontWeight.bold,
          ),
          headline4: TextStyle(
            fontSize: 12.0,
            color: Colors.grey[300],
            fontWeight: FontWeight.w500,
            letterSpacing: 2.0,
          ),
          bodyText1: TextStyle(
            color: Colors.grey[300],
            fontSize: 14.0,
            fontWeight: FontWeight.w600,
            letterSpacing: 1.0,
          ),
          bodyText2: TextStyle(
            color: Colors.grey[300],
            letterSpacing: 1.0,
          ),
        ),
      ),
      home: Builder( // <- Here
        builder: (context) {
          return Column(
            children: [
              Expanded(
                child: Container(
                  color: Theme.of(context).primaryColor,
                ),
              )
            ],
          );
        }
      ),
    );
  }
}