This appears to be a glitch. Sometimes, due to various network problems and spikes, apps behave differently even they're bug free or freshly installed. The only solution I recommend is removing yourself from both accounts and then adding the username back and checking the status. You might need to approve this operation or check the settings to see who can send a friend request or who can add you.
It's also possible that the friend removed you and denied your request to add him/her back.