Вам нужно использовать разные идентификаторы Broadcast для ожидающих намерений.Как то так:
public void startAlarm(int hour, int minute) {
final int _id = (int) System.currentTimeMillis();
Toast.makeText(getApplicationContext(), "Starting Alarm", Toast.LENGTH_SHORT).show();
Calendar calendar = Calendar.getInstance();
calendar.set(Calendar.HOUR_OF_DAY, hour);
calendar.set(Calendar.MINUTE, minute);
calendar.set(Calendar.SECOND, 0);
AlarmManager alarmManager = (AlarmManager) getSystemService(Context.ALARM_SERVICE);
Intent myIntent = new Intent(MainActivity.this, AlarmReceiver.class);
pendingIntent = PendingIntent.getBroadcast(MainActivity.this, _id, myIntent, PendingIntent.FLAG_ONE_SHOT);
alarmManager.setInexactRepeating(AlarmManager.RTC_WAKEUP, calendar.getTimeInMillis(),
AlarmManager.INTERVAL_DAY, pendingIntent);
}