Robert,
I had made serial Y cables from monitoring communications into 2 rs232 ports on a single computer to log data going both directions. Also a Y cable for connecting both a serial printer and scanner to the same RS232 port on a computer.

What you are looking for is along the same lines. RS232 doesn't have a problem sharing an output or splitting send and receive to 2 different endpoints. RS232 ports won't break if 2 outputs are wired funny and connected together. It just won't transfer data.

I expect the printing computer(device) is a legacy device and can't be touched. You can use a custom serial Y cable on the printer.
You need pin 2 and 5 to come from the printer to the monitoring computer.
Then pin 3 and 5 to the legacy computer that is printing. The printing computer may also need the other control pins so the printing application knows the printer is ready.

I found this on amazon.
SF Cable: DB9 Male to 2 Female Serial Rs232 Splitter Cable 12 Inches
If you have a volt meter you can test the wiring.
Otherwise I would use pliers and break all the pins except 2 and 5 on one leg and name it monitor. Then break off only pin 2 on the other leg and and name it printing computer.
If it doesn't work and you ordered 2 cables, the substitute pin 3 in place of pin 2 in the above....
oops version if it doesn't work.. I would use pliers and break all the pins except 3 and 5 on one leg and name it monitor. Then break off only pin 3 on the other leg and and name it printing computer.

You post was from a long time ago but thought maybe it could help.

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Your other choice is to have the legacy computer connected to the monitor computer via RS232 cable. Then the monitor computer connected to the printer. Your application would pass through what it receives in port 1 out to port 2. Then log what it received back from port 2 with the RFID tag information.  In this situation it could store the label information being printed along with the RFID information.
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Another choice is to use a  $20 microcontroller connected as a usb keyboard.The RS232 would be monitored by it and it would be entered into the keyboard.  This may make it possible for the legacy program to actually store the RFID tag data.
Tim

1)     Use RFID labels with a unique identifier already encoded on the label so that we don't need to encode the labels when printed
Sure, we can do that by treating the RFID tag enabled labels as regular labels. But what's the benefit of using an RFID enabled printer to printer labels on RFID tag enabled labels?

2)     As the labels are printed from Computer 1, can we connect Computer 2 at the same time to the printer using another connection (Serial?)
A serial port connection is kind of one-to-one connection. We cannot connect a printer to multiple computers over serial connection. If we want to have 1-to-many connection, we can connect a printer to the network. Then multiple computers on the network can connect to the same printer.

3)     When Computer 1 sends a job to print 5 labels, Is it possible to read the RFID tags for these 5 labels as these are printed and send this to Computer 2 so that I can see the unique identifiers of the RFID Tags
Yes, it's possible. The ZPL Programming Guide has documented how to read the RFID tag through the SGD commands. If the computer 1 is used to send the printer commands to enable the RFID tags, there is no need to read the tag back, because Zebra RFID printer guarantees the encoded data on the tags matches with the data to be encoded. If they don't match, a void label will be printed.