Yup. The 60's show did a lot to establish who would remain relevant in Batman's rogues gallery for years to come. Burgess Meredith did a lot to define the character in pop culture.
When it comes down to the Penguin, you have to realize he's a character with an easily identifiable design and gimmick. All you need to do is give the guy a tux, a trick umbrella, monocle, or some combination of the above.. In recent years, more and more writers have fleshed out the Cobblepott and Wayne connection that Batman and the Penguin have from childhood, making him an enemy that hates both Batman and Bruce Wayne separately, unaware that they're the same person.
He's also compelling because he's often written as this sort of underdog character, whether due to his deformities and handicaps or social status. As other have pointed out, he's the perfect bridge between the insane freaks and the ruthless gangsters of Gotham. He often goes to extreme lengths to prove himself as one of the greats and elites of Gotham. Look at Batman Returns and Gotham, or that one episode of BTAS where he fell in love. You often find yourself rooting for the guy to win against all odds despite how evil he can be. Of course he can also be written as the most disgusting and monstrous man alive which somehow makes him even more fascinating.
He's definitely one of my favorites for sure.