KC Arcade Research
This page contains all the research on the Arcade version that various members has done.
Amie Cards
Amie cards are confirmed to be a Mifare Classic 1K NFC card. This means that they will make NFC enabled phones chime if placed near a NFC reader chip in the phone.
One could potentially use an Arduino to dump the sector data of the card with a NFC reader. You would have to be extremely careful not to overwrite any data bytes stored on the card, or the card might be damaged and/or cause a tampering detected error on the arcade machine.
Kanmusu Cards
The arcade machine can "recongize" the Kanmusu card inserted, but it is unknown if the Kanmusu Cards are RFID or NFC. If they are indeed RFID or NFC, it could be they are not a standard frequency card.
Update 3 August 2017: According to a user from the KanColle Kai Vita Discord, the Kanmusu Cards do indeed use RFID technology. However, it seems to be a case that they might only be "triggered" by a special mechanism. He was also able to confirm that you could potentially stack up to 36 cards on top of each other and the game would have no problem handling it. From what the user said, he was able to get the machine to read 20 cards in one batch.
Also according to the user, the cards that are spat out are blank when they are installed into the machine. The machine would print the character art onto the card and then spit it out. Citation needed.
Part Numbers
One half of the battle is to find out the arcade's parts. If it's possible to get the part numbers of the arcade machine internals, then it would be easier to work out what exactly the machine has inside (ie. the actual IDs of the NFC reader chips, etc).
Assistance Required, Out of Paper
Sometimes the arcade machine can throw this error if it runs into a situation where it has no more paper (or cards?) left in its supply.
An arcade tendant would have then service the machine, install a new batch of paper/cards, and tell the machine to continue.