Each python kernel is separate environment.
If you have installed "azure-keyvault" and "azure-identity" in python 3.8 kernel, it won't be accessible in python 3.10 sdk v2 or another kernel.
You have to install them again if you are switching to different kernel for first time
Please make sure that you are installing with below pip command
%pip install azure-keyvault azure-identity
instead of
!pip install azure-keyvault azure-identity #throws module not found error.
Attached screenshots of trials with different kernels and above command for reference.
Hope it addresses your package installation issues.
Thank you