Why use a date-dependent substring of a pre-shared key instead of just the key?
It is definitely not an industry standard solution. It looks like what the system designer wants to enforce is changing the key everyday. Essentially, it is a naive (and not secure) session key scheme such that each session lasts a day.

The idea of using session key is quite common in secure communication. Cryptanalysis becomes easier if the attacker can have more ciphertexts encrypted with a specific key. Thus often we want to limit the amount of data encrypted under a particular key. This is done by key exchange/establishment protocols that generate random, short-lived keys and secure distribute to sender/receiver. The random keys are called session keys because they are only used in one session. The security of the distribution usually requires e.g. a pre-shared key or asymmetric keys.

Going back to the scheme you mentioned, it is not secure because of many design flaws, e.g. the session keys are not randomly generated, are related to the pre-shared key, and can be re-used. It does not provide much additional security, e.g. if the attacker possess the pre-shared key, he can enumerate the session keys without much effort. It may even weaken the security of the system.