Tidy data in panda dataframe
TL;DR: Array-tidy is extending tidiness across multiple data frames that are indexed by an additional shared dimension, such as time.

From what I understand, array-tidy and dataframe-tidy refer to the underlying data models that are used to achieve your tidy data. It seems to be a reference to how plyr and pandas differ in their data structure logic

Pandas was developed around the idea of panel data, which is your typical two dimensional data frame of observations and variables, but with samples accumulated over a period of time. Pandas name even stems from this particular type of data (panel data structures, per this McKinney presentation and the pandas docs). The inclusion of a third dimension, thinking about your data over time, seems to have informed some thinking around how pandas would evolve. I recommend checking out the slides (specifically the one titled "The pandas killer feature: indexing"). 

Relating this back to the Tidy Data paper, creating multiple data frames across an additional dimension (like you would with a Panel in prior versions of pandas) seems to qualify as the "high-dimensional arrays" Wickham discusses in the section you quoted. 

I hope this helps!