Yes, with the Stocard account system 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 registered 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 registration. If you register 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.
If you would like to share your Stocard account with family members, please make sure both of you have already registered their own account before synchronizing the devices. This way, you can ensure that it is always possible to switch back to an independent account in case you want to separate devices again.