The official description claims that this is a younger and "unrefined" Batman facing a defining moment in his career as a crime-fighter. That harkens back to
In Year One, Batman roots out mob rule, but creates more dangerous adversaries
The story uses it to reflect on the central theme of the novel, that any man is "one bad day" from turning evil. In the context of an Arkham prequel, such a story could also be used to illustrate Batman's inexperience in his early years, and to tie thematically to Joker's fate at the end of Arkham City. It's worth noting, though, that this is only one of Joker's many origin stories that may be outright fabrications, and the character has said he prefers his background to be "multiple choice."
Rumors circulating around the Silver Age influence are more complex. The Silver Age of comics lasted through the late 50s and 60s, and was heavily influenced by the Comics Code Authority. Citing concerns of adult content in books that were primarily read by children, publishers voluntarily restricted their content to kid-friendly stories with strict limits on depictions of violence, drugs, and sex.
The Joker giving Commissioner Gordon his 'one bad day'
If a character named "Deathstroke" seems incongruous with that, you're not wrong. The character was introduced in 1980, a decade after the Silver Age had ended. At this point comics were more splashy and on the verge of becoming so hyper-violent that it was satirized in 1987's
Deathstroke introduces some of the old utraviolence