One of the biggest departures from their more traditional depictions of the usual Batman cast was that of Caped Crusader's version of the Riddler, thus named in-universe because he "riddles" everything with blunts (although he does at least once do the usual riddle/joke schtick).
While a large departure from most versions of the character, Caped Crusader's Eddie is actually intended to be a specific meta joke, which like a lot of stuff in season two, kinda relies on the audience's familiarity with early/mid-century crime cinema (or at least the curiousity to look it up afterwards).
In this case, Eddie's appearance, personality and trigger-happy violence is intended as a reference to the character Tommy Udo, as played by Richard Widmark in the 1947 film noir Kiss of Death...
...With Widmark's performance as Udo being a key inspiration for Frank Gorshin's depiction of the Riddler in the 1960s live-action Batman show with Adam West.
So, rather than being entirely random, Caped Crusader's version of the Riddler is instead a call-back to the inspiration of an earlier version of the character rather than, say, the Batman drawing parallels with the Zodiac Killer or something.
Again, it makes sense, it just kinda hinges on the audience's patience to see what they're doing?