Yes, with a Stocard account it is indeed possible to synchronize your cards, passes and points balances across multiple devices. In order to synchronize your data, simply log into the same Stocard account on all of your devices.
Keep in mind to use your main device (the device where all your cards are stored) for the account sign up. If you sign up for your account on a device that has no cards added yet, there won't be any cards in your Stocard account.
Please also note, that if you synchronize your devices, deleting a card on one of your devices will lead to the card getting deleted on all connected devices and in your Stocard account as well.