How is the usual way of using PKCS5/7 when a weird data is given?
The PKCS 5/7 encoder would not do anything since it has all 16 bytes (1 block size required by AES cipher).


That's where you're wrong. 

Padding should be always applied. If the user is encrypting a plaintext that is already a multiple of the block size, then the PKCS#5/#7 padding schemes will add an additional full block of padding, so that there is no ambiguity when decrypting.

In your example, that would mean this is the padded plaintext that is encrypted:

05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10


There are of course padding schemes that don't add an additional block (for example PHP's zero padding), but that's not how PKCS#5 or PKCS#7 works.