All of this information is available on the partner hub but there seems to be a general confusion regarding it.
HOW TO GET TO "MY PARTNER INFO":
https://partner.starbucks.com/mpi/Pages/globalMPI.aspx
Scroll down and select "Log into My Partner Info from a personal device (view only) >>".
Log in using only numbers and password. Do not put US or CA in front. Click on the Partner Card link. This will provide a 16-digit card number, AND an 8-digit security code for use in the app. If you would like a plastic card, it can be requested here.
STARBUCKS APP:
Go to cards, scroll to the bottom. Click on Add Card and enter appropriate numbers. It should recognise them as a partner card.
FOR MOBILE DISCOUNT:
https://www.starbucks.com/account/signin
Go to the account page. You might need to turn on desktop mode if you are using a phone. At the bottom of the "Personal Info" link, click on "Add your partner number." and follow the instructions.
NOTE: It might take a day or two to update in the app.