If we allow a database, what complexity class it is?
You can say anything you want, if you define your terms and your notation and make it clear what you are saying.

In this case I would not expect someone to know what you mean by $M/f(n)$ or by $A \in M/f(n)$.  Something like $P/\text{poly}$ is well-understood, but $P$ is a complexity class, and $M$ is a specific machine.  Rather than risking confusion, just say what you mean -- it only takes a sentence to say that the problem can be decided by $M$ using advice of size $f(n)$.