Pure speculation time going off the following picture.
Let us assume that this shows all the data on a Amiibo for Smash Bros. Let us also assume that the "new fighting tactics" that Amiibo's learn while fighting are actually dynamically calculated from the type (Speedster) and Attack/Defense/Speed stats.
So!
Level - Known to be capped at 50. 6 bits.
Name - I don't see a way to edit the name so lets assume this is not possible
Type - Not known how many types there are. Assuming 15 so 4 bits
Stats - The attack-stat seems to be about half way the bar so it can be assumed that they cap at 255. 3 * 8 bit = 24 bit
Effects - Not known how many effects there are so let's assume 255. 3 * 8 bit = 24 bit
Specials - Think there are 3 different moves for every slot (correct me if I'm wrong) so 4 * 2 bit = 8 bit
6 + 4 + 24 + 24 + 8 = 66 bits = 8.25 byte.
There probably is more information being stored so let us double it to 16 byte needed for Smash Bros.
Looking at the storage capabilities of NFC tags
http://www.radio-electronics.com/info/wireless/nfc/near-field-communications-tags-types.php
Type 1: 96 byte - 2KByte
Type 2: 48 byte - 2KByte
Type 3: 2KByte
Type 4: up to 32Kbyte
I do not know which type the WiiU Pad / new3DS support but I think it is safe to assume that they cheaped out and went either for a type 1 or 2. Also assuming that they aren't complete assholes and actually go to the limit with storage capacity, I don't see having 1 Amiibo for a ton of different games being a problem.
Then again I could just as well be COMPLETELY wrong.