Unfortunately the document is not freely available. Can you elaborate how it works?

The latest standard (2019) is divided into two parts: ISO 8601-1 and ISO 8601-2. Part 2 is titled "Extensions" and includes many different - but optional - ways to extend representations of dates and times. In my opinion, Part 2 represents the "leftovers" - that which was too domain specific or esoteric to warrant broad implementation.

Representations of seasons, quarters, etc. is one of many types of extensions included in Part 2.

These extensions are always used in "YYYY-MM" format (and always include the "-"). For the "MM" field, you use the numbers 21-41 inclusive. The numbers represent, in order, spring, summer, autumn, and winter, in 3 ways: independent of hemisphere (21-24), northern hemisphere specifically (25-28), southern hemisphere specifically (29-32); quarters 1-4 (33-36), quadrimesters 1-3 (37-39), and finally, semestrals 1-2 (40-41).

This, in combination with the fact that the lookup table is paywalled, makes it a great competitor for the "extremely pointlessly overengineered standards" award.

I wonder if there is any software that actually supports that.

Where's the 33 come from?

There is a list of "Sub-year groupings" which assign two-digit numbers to seasons (hemisphere independent), seasons (Northern Hemisphere), seasons (Southern Hemisphere), quarters, quadrimesters (four-month periods), and semestrals (six-month periods). The numbering starts at 21.

The numbers 1 to 12 are already used to represent months in formats such as YYYY-MM, so starting this list higher than 12 allows these (certainly rare) representations to coexist. e.g. "2021-12" represents "October, 2021" and "2021-29" represents "Spring, Southern Hemisphere, 2021".