Using string.equals() and not getting the results...

Projects Programming
3 
  while(rfid.available())
  {
    if(rfid.read() == 255)
    {
      for(int i=1;i<11;i++)
      {
        Tag[i]= rfid.read();
      }
    }
  }
}

So, if there is one byte available to read, read all 12 of them. Got it. Perhaps you need to rework this bit of rubbish.

void loop()
{
  read_serial();
  
}

Now, it's a great idea to use functions, but this is ridiculous.

    char buffer[1];
    for(int index = 8; index>4; index--)
    {
      itoa(Tag[index], buffer,16);

Tag[n] contains a byte, in the range 0 to 255. Converting that to a hex string will result in a string between "00" to "FF". Even if the leading 0 is suppressed, the trailing NULL that itoa() adds means that the array size needs to be at least 3 elements, not 1.

      TagID = TagID += buffer;

Waste, waste, waste. You REALLY need to figure out exactly what this statement is doing. You probably will not believe the amount of resources this simple (mangled) statement consumes.