As of Keyring 15.2.0, you can use keyring.get_credential to store a username and password together. The original answer has been kept below for reference.
While keyring was only designed to store passwords, you can abuse get_password to store the username separately.
import keyring
# store username & password
keyring.set_password("name_of_app", "username", "user123")
keyring.set_password("name_of_app", "password", "pass123")
# retrieve username & password
username = keyring.get_password("name_of_app", "username")
password = keyring.get_password("name_of_app", "password")
Alternatively, if you want to keep the username paired with the password:
import keyring
service_id = "name_of_app"
username = "user123"
# store username & password
keyring.set_password(service_id, "username", username)
keyring.set_password(service_id, username, "pass123")
# retrieve username & password
username = keyring.get_password(service_id, "username")
password = keyring.get_password(service_id, username)
Credit to Dustin Wyatt & Alex Chan for this solution.