As far as I know (and I used MTM) about MTM, it seems not feasible with the record and playback feature of MTM. This is just a simple recorder which has been designed and provided to help manual tester to ease and accelerate their job of manual testing. All validations (Pass/Fail) still needs to done manually by the tester.
If you want to go further in testing using MS products, then you need to enter the rabbit hole and start using Coded UI, your first level of analysis of MTM can be helpful in creating and using Coded UI tests, as you can convert your recorded MTM test cases into Coded UI code and then can manipulate them for any level of testing (like Copy and retain a generated value and passing it to another page -- which is your required functionality). So, you nee
In addition to support above mentioned text I found this:
The record and playback in Microsoft Test Manager is designed to
assist the manual tester, it is not a complete automation solution. It
helps to make retesting easier and more precise. All validation is
done manually in Test Manager, in saying that - Test Manager records
when you input Parameters so this means that it is easier to convert
these recordings to a Coded UI Test. This enables programmatic
validations and you can do more complex scenarios as well.
So Microsoft Test Manager is focused around helping the manual testers
Coded UI tests are performed in Visual Studio and can be created from
a recording in Test Manager and is intended for a specialist type
role, normally you would want a bit of programming knowledge. bit it
would still be possible to create basic tests with simple validations
using the recording tool in visual studio
Thus, you will need to start using Coded UI, for achieving these kinds of mentioned scenarios.