For output coupling caps, the rated ripple current is an important factor of interest because it's current being pumped to the speakers, and phase shift to keep best linearity across the audible bandwidth. So look at the rated ripple current between type I, II, III ( well, spoiler alert.... it becomes better when increasing type number....). But it's handy when comparing different brands.
Phase shift becomes less with increased rated voltage (that applies for any electrolytic capacitor). So for that reason, going to 100V rated caps will provide less phase shift. And type III has the least phase shift between the types for a given rated voltage.
ESR is no interest for this application. However, ESL (inductivity) is very much of interest due to the current application, and output stages don't like inductive loads
This is where audio caps like the FG and KG come in, their ESL is nicely low (due to the way they are winded and internal termination to the external terminals).
ESL is rarely mentioned on datasheets, one of the reasons why I invested in serious RCL testgear with measuring frequencies covering 0 to 200kHz, and sharing the obtained knowledge here.
The reason why the casing size increases from type I to III is due to the thicker insulating film between layers.
And as the Japanese domestic high end DIY audio scene is still very much hardwired tube amp orientated, and Nichicon being deeply involved in audio parts, they (luckily) maintain a solder lug version.